A Carnot heat engine with thermal efficiency 110110 is run backward as a Carnot refrigerator.
What is the refrigerator's coefficient of performance? Express your answer using one significant figure.

Part (a) The expression for the image distance, d1 is di = 23.4 cm

Part (b) )Numerically, the distance of the image, d1 is 23.4 cm.

d0 = 19 cm

focal length is f = 10.5 cm.

The formula used for the distance of the image for a concave mirror is given as follows:

1/f = 1/do + 1/di

Where,

f = focal length

do = object distance from the mirror, and

di = image distance from the mirror

Part (a)

we substitute the given values in the above formula.

1/10.5 = 1/19 + 1/di

Multiplying both sides by 10.5 × 19 × di, we get:

19 × di = 10.5 × di + 10.5 × 19

Subtracting 10.5 from both sides, we get:

19 × di - 10.5 × di = 10.5 × 19

Combining like terms, we get:

di(19 - 10.5) = 10.5 × 19

Dividing both sides by (19 - 10.5), we get:

di = 10.5 × 19/(19 - 10.5)

di = 10.5 × 19/8.5

di = 23.4 cm

Therefore, the expression for the image distance, d1 is di = 23.4 cm

Part (b)

Numerically, the distance of the image, d1 is 23.4 cm.

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Suppose that the separation between two speakers A and B is 6.70 m and the speakers are vibrating in-phase. he largest possible distance between speaker B and the observer, such that destructive interference is observed, is 1.62 meters.

To observe destructive interference, the path difference between the waves reaching the observer from speakers A and B must be a multiple of half the wavelength. In this case, the frequency of the tone is 101 Hz, corresponding to a wavelength of λ = (speed of sound / frequency) = 3.39 m.

Since the observer is directly facing speaker B and the line connecting A and B is perpendicular to the observer's line of sight, the path difference is simply the difference in distance traveled by the waves from A and B to the observer.

Let's assume that the distance between speaker B and the observer is x. Then, the path difference can be expressed as follows:

Path difference = distance AB - distance AO = 6.70 m - x

For destructive interference, the path difference must be (n + 1/2)λ, where n is an integer. So, we have:

6.70 m - x = (n + 1/2) * 3.39 m

Simplifying the equation, we can solve for x:

x = 6.70 m - (n + 1/2) * 3.39 m

The largest possible distance between speaker B and the observer occurs when n is the smallest positive integer that satisfies the equation. In this case, n = 1, giving:

x = 6.70 m - (1 + 1/2) * 3.39 m = 6.70 m - 5.08 m = 1.62 m

Therefore, the largest possible distance between speaker B and the observer, such that destructive interference is observed, is 1.62 meters.

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A proton (mass m = 1.67 x 10⁻²⁷ kg) is being accelerated along a straight line at 5.30 x 10¹¹ m/s2 in a machine. If the proton has an initial speed of 9.70 x 10⁴ m/s and travels 3.50 cm, (a)The speed of the proton is approximately 6.125 x 10⁵ m/s.(b) The increase in kinetic energy is approximately 1.87 x 10⁻¹⁸ Joules.

(a) To find the final speed of the proton, we can use the equation:

v² = u² + 2as

Where:

v = final velocity

u = initial velocity

a = acceleration

s = displacement

Plugging in the given values:

u = 9.70 x 10⁴ m/s

a = 5.30 x 10¹¹ m/s²

s = 3.50 cm = 3.50 x 10⁻² m

Calculating:

v² = (9.70 x 10⁴ m/s)² + 2(5.30 x 10¹¹ m/s²)(3.50 x 10⁻² m)

v² = 9.409 x 10⁸ m²/s² + 3.71 x 10¹⁰ m²/s²

v² = 9.409 x 10⁸ m²/s² + 3.71 x 10¹⁰ m²/s²

v² = 3.753 x 10¹⁰ m²/s²

Taking the square root of both sides to find v:

v = √(3.753 x 10¹⁰ m²/s²)

v ≈ 6.125 x 10⁵ m/s

Therefore, the speed of the proton is approximately 6.125 x 10⁵ m/s.

(b) The increase in kinetic energy can be calculated using the equation:

ΔK = (1/2)mv² - (1/2)mu²

Where:

ΔK = change in kinetic energy

m = mass of the proton

v = final velocity

u = initial velocity

Plugging in the given values:

m = 1.67 x 10⁻²⁷ kg

v = 6.125 x 10⁵ m/s

u = 9.70 x 10⁴ m/s

Calculating:

ΔK = (1/2)(1.67 x 10⁻²⁷ kg)(6.125 x 10⁵ m/s)² - (1/2)(1.67 x 10⁻²⁷ kg)(9.70 x 10⁴ m/s)²

ΔK = (1/2)(1.67 x 10⁻²⁷ kg)(3.76 x 10¹¹ m²/s²) - (1/2)(1.67 x 10⁻²⁷ kg)(9.409 x 10⁸ m²/s²

ΔK ≈ 1.87 x 10⁻¹⁸ J

Therefore, the increase in kinetic energy is approximately 1.87 x 10⁻¹⁸ Joules.

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A.5.0 mH inductor is connected in parallel with a variable capacitor.  the minimum oscillation frequency for this circuit is approximately 1.06 MHz. the minimum oscillation frequency for this circuit is approximately 1.06 MHz.

To determine the minimum and maximum oscillation frequencies for the circuit consisting of a 5.0 mH inductor and a variable capacitor ranging from 140 pF to 380 pF, we can use the formula for the resonant frequency of an LC circuit:

f = 1 / (2π√(LC))

The resonant frequency, f, is the frequency at which the circuit exhibits maximum oscillation or resonance. The minimum oscillation frequency occurs when the capacitance is at its maximum value, and the maximum oscillation frequency occurs when the capacitance is at its minimum value.

For the minimum oscillation frequency:

C = 380 pF = 380 × 10^(-12) F

L = 5.0 mH = 5.0 × 10^(-3) H

Substituting these values into the formula, we get:

f_min = 1 / (2π√(5.0 × 10^(-3) H × 380 × 10^(-12) F))

     = 1 / (2π√(1.9 × 10^(-15) H·F))

     ≈ 1.06 MHz

Therefore, the minimum oscillation frequency for this circuit is approximately 1.06 MHz.

For the maximum oscillation frequency, we use the minimum value of the capacitor:

C = 140 pF = 140 × 10^(-12) F

Substituting this value into the formula, we get:

f_max = 1 / (2π√(5.0 × 10^(-3) H × 140 × 10^(-12) F))

     = 1 / (2π√(7.0 × 10^(-16) H·F))

     ≈ 2.04 MHz

Therefore,  the minimum oscillation frequency for this circuit is approximately 1.06 MHz.

In summary, the minimum oscillation frequency is approximately 1.06 MHz, occurring when the capacitor is at its maximum value of 380 pF. The maximum oscillation frequency is approximately 2.04 MHz, occurring when the capacitor is at its minimum value of 140 pF. These frequencies represent the resonant frequencies at which the LC circuit will exhibit maximum oscillation or resonance.

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When an object is dropped close to Earth's surface, it undergoes free fall motion. It accelerates downward due to gravity, gaining speed as it falls. However, in the absence of air resistance, the object will continue to accelerate until it hits the ground or another surface.

When an object is dropped close to Earth's surface, it experiences the force of gravity pulling it downward. Gravity is an attractive force between two objects with mass, in this case, the object and the Earth. The acceleration due to gravity near the Earth's surface is approximately 9.8 m/s², denoted by the symbol 'g'.

As the object is released, it initially has an initial velocity of 0 m/s because it is not moving. However, as it falls, it accelerates downward due to gravity. The object's velocity increases over time as it gains speed. The acceleration is constant, so the object's velocity changes at a steady rate.

The motion of the object can be described by the equations of motion. The displacement (distance) covered by the object is given by the formula s = ut + (1/2)gt², where s is the displacement, u is the initial velocity, t is the time, and g is the acceleration due to gravity.

Additionally, the velocity of the object can be determined using the equation v = u + gt, where v is the final velocity.

During free fall, the object continues to accelerate until it reaches its maximum velocity when air resistance becomes significant. However, in the absence of air resistance, the object will continue to accelerate until it hits the ground or another surface.

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a) The entropy can be written as S = -kΣ Pr ln Pr, where Pr is the probability of observing the system in state r with energy Er.

b) The mean Helmholtz free energy is related to the partition function according to Z = exp(-βF).

a) To demonstrate this, we start with the definition of entropy:

S = -kΣ Pr ln Pr.

We substitute

Pr = exp(-βEr)Z into the equation,

where β = 1/(kT) and Z = Σ exp(-βEr) is the partition function.

After substitution, we have

S = -kΣ (exp(-βEr)Z) ln (exp(-βEr)Z).

By rearranging terms and simplifying, we obtain

S = -kΣ (exp(-βEr)Z) (-βEr - ln Z).

Further simplification leads to S = kβΣ (exp(-βEr)Er) + kln Z, and since

β = 1/(kT), we have S = Σ PrEr + kln Z.

Finally, using the definition of mean energy as

U = Σ PrEr, we arrive at

S = U + kln Z, which is the expression for entropy.

b) To demonstrate this, we start with the definition of Helmholtz free energy:

F = -kTlnZ.

We rewrite this equation as

lnZ = -βF.

Taking the exponential of both sides, we obtain

exp(lnZ) = exp(-βF),

which simplifies to

Z = exp(-βF).

Therefore, the mean Helmholtz free energy is related to the partition function by Z = exp(-βF).

These relationships demonstrate the connections between entropy, probability distribution, partition function, and mean Helmholtz free energy in a system in thermal equilibrium with a heat bath at temperature T. The canonical probability distribution and partition function play crucial roles in characterizing the statistical behavior and thermodynamic properties of the system.

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The statement "The magnitude of a vector quantity is considered a scalar quantity" is NOT true. The magnitude of a vector represents its size or length and is always a scalar quantity

A scalar quantity only has magnitude and no direction. On the other hand, a vector quantity includes both magnitude and direction. Therefore, the magnitude of a vector cannot be considered a scalar quantity.

Regarding the given directions, "Go North 10 miles, then East 4 miles, and then South 7 miles," we can calculate the distance from the starting position by considering the net displacement. Moving North 10 miles and then South 7 miles cancels out the vertical displacement, resulting in a net displacement of 3 miles to the North.

Moving East 4 miles adds to the net displacement, giving us a final displacement of 3 miles North and 4 miles East. By using the Pythagorean theorem, the distance from the starting position is calculated as [tex]\sqrt(3^2 + 4^2) = \sqrt(9 + 16) = \sqrt25 = 5[/tex] miles.

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The true statement concerning a convex mirror is: A convex mirror produces a smaller image than a plane mirror does for the same object distance.

A convex mirror is a curved mirror that bulges outward. It has a reflective surface that curves away from the incident light. Due to its shape, a convex mirror diverges light rays and forms a virtual image. The image formed by a convex mirror is always upright (not inverted) and smaller in size compared to the object. This is why the statement "A convex mirror produces a smaller image than a plane mirror does for the same object distance" is true.

In contrast, a plane mirror produces an image that is the same size as the object and has no distortion or magnification. When light rays from an object fall on a convex mirror, they reflect in a way that diverges the rays, causing the image to appear smaller than the actual object. This reduction in size is a result of the way the convex mirror curves and reflects light.

The curved shape of a convex mirror is not necessarily required to be perfectly spherical. While many convex mirrors do have a spherical shape, there can be variations in the curvature depending on the specific design and purpose of the mirror.

Additionally, a convex mirror forms virtual images, which means the image cannot be projected onto a screen. Virtual images are formed by the apparent intersection of the reflected light rays, and they are always located behind the mirror. Therefore, a convex mirror cannot form a real image.

In summary, the statement "A convex mirror produces a smaller image than a plane mirror does for the same object distance" is true. The curved shape of a convex mirror and its ability to diverge light rays result in a virtual image that is smaller and upright compared to the object.

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(a) the mass of the rod is [tex]$7.0 * 10^{-8}kg$[/tex].

(b) the potential energy change that occurs in [tex]$0.205s$[/tex] is [tex]$8.8 * 10^{-21}J$[/tex].

(c) the electrical energy dissipated in the resistor in [tex]$0.20s$[/tex] is [tex]$4.6 * 10^{-21}J$[/tex].

(a) Mass of the rod

The magnetic force acting on the rod causes a component of the gravitational force to be balanced. The component is that which pulls the rod downwards along the track. Therefore, the magnetic force acting on the rod is equal in magnitude but opposite in direction to the component of the gravitational force. Since the force is perpendicular to the velocity of the rod, it does not do any work. This implies that the kinetic energy of the rod is constant. This gives us the equation of motion of the rod as,

[tex]$mg\sinθ = BIl$[/tex]

[tex]$mg\sinθ = Bvq$[/tex]

Where the [tex]$v$[/tex] is the speed of the rod. Since the resistance of the rod and tracks is negligible, the potential difference between the points A and B is zero. This means that the electrical potential energy lost by the rod is equal to the gravitational potential energy gained by the rod. Therefore, [tex]$mgΔh = qvB$l[/tex]

where [tex]$\Delta h$[/tex] is the vertical distance through which the rod falls. Since [tex]$l=1.8m$, $\sinθ = \frac{1}{\sqrt{1+4/9}} ≈ 0.74$[/tex]. Thus,

[tex]$m = \frac{qBvl}{g\sin\theta}$[/tex]

Substituting the given values, we get,

[tex]$m = \frac{(1.6 * 10^{-19})(0.57)(4)(1.8)}{(9.8)(0.74)}$[/tex]

Therefore, the answer is [tex]$7.0 * 10^{-8}kg$[/tex].

Part (b)The potential energy lost by the rod when it drops a distance $\Delta h$ is given by,

[tex]$mg\delta h = qvB$l[/tex]

Thus, the potential energy change in a time of [tex]$0.205s$[/tex] is,

[tex]$\Delta U = mg\Deltah\frac{\Delta t}{v} = \frac{qB\Delta h}{v}$[/tex]

Substituting the given values, we get,

[tex]$\Delta U = \frac{(1.6 * 10^{-19})(0.57)(0.205)}{4}$[/tex]

Therefore, the answer is [tex]$8.8 * 10^{-21}J$[/tex].

Part (c)The electrical energy dissipated in the resistor is equal to the change in the potential energy of the rod, i.e. the gravitational potential energy lost by the rod. This is given by,

[tex]$\Delta U = mg\Delta h = qvB$l[/tex]

where [tex]$\Delta h[/tex]$ is the vertical distance through which the rod falls. Substituting the given values, we get,

[tex]$\Delta U = \frac{(1.6 * 10^{-19})(0.57)(0.20)}{4}$[/tex]

Therefore, the answer is [tex]$4.6 * 10^{-21}J$[/tex].

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The most positive output level is +2VI, and the most negative output level is -2VI.

The input and output waveforms of the given circuits are shown below:

Part (b) - Input voltage = VI

The diode in this circuit is forward-biased, so it conducts and limits the output voltage to +0.7 V. Therefore, the output waveform is a constant +0.7 V.

Part (c) - Input voltage = V

In this circuit, both diodes are reverse-biased, so they do not conduct. Therefore, the output waveform is a constant 0 V.Part

(d) - Input voltage = VI

This circuit is a voltage doubler. During the first half-cycle, the input voltage charges capacitor C1 to VI. In the second half-cycle, the bottom diode is forward-biased, and the top diode is reverse-biased. As a result, the output voltage is equal to twice the voltage across capacitor C1. The output voltage is therefore +2VI during the second half-cycle. During the next half-cycle, the output voltage is -VI because the input voltage is -VI, and the output voltage cannot change instantaneously. During the fourth half-cycle, the output voltage is -2VI.

Therefore, the output waveform is a square wave with an amplitude of 2VI and a duty cycle of 0.5. The most positive output level is +2VI, and the most negative output level is -2VI.

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A 2.2-kg block is released from rest at the top of a frictionless incline that makes an angle of 40° with the horizontal.  the maximum distance the block can compress the spring is approximately 0.181 m.

To find the maximum distance the block can compress the spring, we need to consider the conservation of mechanical energy.

The block starts from rest at the top of the incline, so its initial potential energy is given by mgh, where m is the mass of the block, g is the acceleration due to gravity, and h is the height of the incline. The height h can be calculated using the angle of the incline and the distance L:

h = L*sin(40°)

Next, we need to find the final potential energy of the block-spring system when the block compresses the spring to its maximum extent. At this point, all of the block's initial potential energy is converted into elastic potential energy stored in the compressed spring:

0.5kx^2 = mgh

Where k is the spring constant and x is the maximum compression distance.

Solving for x, we have:

x = sqrt((2mgh) / k)

Substituting the given values:

x = sqrt((2 * 2.2 kg * 9.8 m/s^2 * L * sin(40°)) / 280 N/m)

Calculating the value:

x ≈ 0.181 m

Therefore, the maximum distance the block can compress the spring is approximately 0.181 m.

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The separation of the slits is approximately 6.68 × 10^-4 mm.

The separation of the slits can be determined using the formula for interference maxima. In this case, the separation of the interference maxima on the screen and the distance between the screen and the slits are given, allowing us to calculate the separation of the slits.

In interference experiments with double slits, the separation between the slits (d) can be determined using the formula:

d = (λ * L) / (m * D)

where λ is the wavelength of light, L is the distance between the slits and the screen, m is the order of the interference maximum, and D is the separation between consecutive interference maxima on the screen.

In this case, the wavelength of light is given as 539.1 nm (or 5.391 × 10^-4 mm), the distance between the slits and the screen (L) is 0.84 m (or 840 mm), and the separation between consecutive interference maxima on the screen (D) is given as 1.04 mm.

To find the separation of the slits (d), we need to determine the order of the interference maximum (m). The order can be calculated using the relationship:

m = D / d

Rearranging the formula, we have:

d = D / m

Substituting the given values, we find:

d = 1.04 mm / (840 mm / 5.391 × 10^-4 mm) ≈ 6.68 × 10^-4 mm

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The arranged statements based on series are: As warm moist air is less dense than cooler air, it begins to rise, Air moves up in altitude, and the temperature of air decreases.

Thus, air pressure at this location is considered low pressure. Therefore, the answer is as follows: D, C, B, and A.

Low-pressure systems are found near the equator, where warm air rises, or in temperate zones. A high-pressure zone is created where cold air sinks. In a low-pressure zone, the air is forced upward, and clouds and precipitation occur.Air pressure at this location is considered low pressure.

As warm moist air is less dense than cooler air, it begins to rise, Air moves up in altitude, and the temperature of air decreases. The reduction in air pressure causes the vapor to cool, and as it cools, the capacity of air to hold vapor decreases until the temperature reaches the dew point.

When this happens, the water vapor condenses into tiny liquid droplets, forming a cloud.Warm, moist air rises until it reaches a point where the temperature drops to the dew point. As it cools, it can no longer hold the same amount of moisture, and the excess moisture forms clouds.

The cloud grows as more water vapor condenses on the surface of the droplets, increasing their size and weight until they fall to the ground as rain, snow, or hail.

The process of the formation of clouds is a fascinating one.

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a) The gravitational field strength near the "surface" of the Sun is approximately 274.7 N/kg b) The semi-major axis length for the fictional comet's orbit is approximately 7.78 × 10^11 meters. c) The material from the coronal mass ejection (CME) would travel approximately 4.14 × 10^8 meters from the center of the Sun before coming to rest due to the Sun's gravity.

a) Gravitational field near the "surface" of the Sun:

Using the formula:

[tex]\[ g = \frac{{G \cdot M}}{{r^2}} \][/tex]

where [tex]\( G \)[/tex] is the gravitational constant, [tex]\( M \)[/tex] is the mass of the Sun, and [tex]\( r \)[/tex] is the radius of the Sun. Substituting the given values, we have:

[tex]\[ g = \frac{{(6.67430 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2) \cdot (1.989 \times 10^{30} \, \text{kg})}}{{(700 \, \text{million meters})^2}} \approx 274.7 \, \text{N/kg} \][/tex]

Therefore, the gravitational field near the "surface" of the Sun is approximately 274.7 N/kg.

b) Semi-major axis length for the fictional comet's orbit:

Using Kepler's third law equation:

[tex]\[ a = \left( \frac{{T^2 \cdot GM}}{{4\pi^2}} \right)^{1/3} \][/tex]

where [tex]\( T \)[/tex]is the orbital period of the comet,[tex]\( G \)[/tex] is the gravitational constant, and [tex]\( M \)[/tex] is the mass of the Sun. Substituting the given values, we get:

[tex]\[ a = \left( \frac{{(100 \, \text{years})^2 \cdot (6.67430 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2) \cdot (1.989 \times 10^{30} \, \text{kg})}}{{4\pi^2}} \right)^{1/3} \approx 7.78 \times 10^{11} \, \text{m} \][/tex]

Therefore, the semi-major axis length for the fictional comet's orbit is approximately [tex]\( 7.78 \times 10^{11} \) meters.[/tex]

c) Distance traveled by material from a coronal mass ejection (CME):

Using the equation:

[tex]\[ r = \frac{{GM}}{{2v^2}} \][/tex]

where [tex]\( G \)[/tex] is the gravitational constant,[tex]\( M \) i[/tex]s the mass of the Sun, and [tex]\( v \)[/tex] is the average speed of the CME. Substituting the given values, we have:

[tex]\[ r = \frac{{(6.67430 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2) \cdot (1.989 \times 10^{30} \, \text{kg})}}{{2 \cdot (550 \, \text{m/s})^2}} \approx 4.14 \times 10^{8} \, \text{m} \][/tex]

Therefore, the material from the coronal mass ejection (CME) would travel approximately [tex]\( 4.14 \times 10^8 \)[/tex]meters from the center of the Sun before coming to rest due to the Sun's gravity.

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The magnitude of the magnetic field must be 1.20 × 10⁻³ T to produce a flux of 3.80 × 10⁻³ Wb through the surface.

The formula to calculate the magnetic flux through a surface is given by,Φ=BAcosθHere,Φ is the magnetic flux, B is the magnetic field, A is the area of the surface, and θ is the angle between the magnetic field and the normal to the surface. Let's solve for part A.

Step 1. Given,Area of the surface, A = 3.75 cm x 3.25 cm = 12.1875 cm². The angle between the magnetic field and the normal to the surface, θ = 25°Magnetic flux through the surface, Φ = 3.80 × 10⁻³ Wb.

Step 2.Substituting the given values in the formula,Φ=BAcosθ⇒B=Φ/(Acosθ)⇒B=3.80×10⁻³/(12.1875×cos 25°)B=1.20 × 10⁻³ TSo, the magnitude of the magnetic field must be 1.20 × 10⁻³ T to produce a flux of 3.80 × 10⁻³ Wb through the surface.

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The very far from the disk, the approximate value of D on the z-axis is zero.

To find the approximate values of D on the z-axis for the given scenarios, we can use appropriate Gaussian surfaces.

a) Very close to the disk (O < z << a):

In this case, we can consider a cylindrical Gaussian surface of radius r and height dz, centered on the z-axis and very close to the disk. The disk lies in the xy-plane, and its charge density is given by ps(r, ϕ).

Using Gauss's law, we have:

∮ D · dA = Q_enclosed

Since the electric field D is radially directed and the Gaussian surface is cylindrical, the dot product D · dA simplifies to D · dA = D(2πr dz).

The enclosed charge Q_enclosed is the charge within the cylindrical Gaussian surface, which is given by:

Q_enclosed = ∫∫ ps(r, ϕ) r dr dϕ

Applying Gauss's law, we get:

D(2πr dz) = ∫∫ ps(r, ϕ) r dr dϕ

Since ps(r, ϕ) is non-uniform, we cannot simplify the integral further. However, in the limit of dz approaching zero, the contribution from ps(r, ϕ) to the integral becomes negligible. Therefore, we can approximate the integral as ps(0, ϕ) multiplied by the area of the disk, which is πa^2:

D(2πr dz) ≈ ps(0, ϕ) πa^2

Dividing both sides by 2πr dz, we get:

D ≈ ps(0, ϕ) πa^2 / (2πr dz)

D ≈ (ps(0, ϕ) a^2) / (2r dz)

Since we are interested in the value of D on the z-axis (r = 0), we have:

D ≈ (ps(0, ϕ) a^2) / (2(0) dz)

D ≈ (ps(0, ϕ) a^2) / 0

As the denominator approaches zero, we can approximate D as:

D ≈ (ps(0, ϕ) a^2) / 0 = ∞

Therefore, very close to the disk, the approximate value of D on the z-axis is infinite.

b) Very far from the disk (z >> a):

In this case, we can consider a cylindrical Gaussian surface of radius R and height dz, centered on the z-axis and very far from the disk. The disk lies in the xy-plane, and its charge density is given by ps(r, ϕ).

Using Gauss's law, we have:

∮ D · dA = Q_enclosed

Since the electric field D is radially directed and the Gaussian surface is cylindrical, the dot product D · dA simplifies to D(2πR dz).

The enclosed charge Q_enclosed is the charge within the cylindrical Gaussian surface, which is given by:

Q_enclosed = ∫∫ ps(r, ϕ) r dr dϕ

Applying Gauss's law, we get:

D(2πR dz) = ∫∫ ps(r, ϕ) r dr dϕ

Similar to the previous case, in the limit of dz approaching zero, the contribution from ps(r, ϕ) to the integral becomes negligible. Therefore, we can approximate the integral as ps(0, ϕ) multiplied by the area of the disk, which is πa^2:

D(2πR dz) ≈ ps(0, ϕ) πa^2

Dividing both sides by 2πR dz, we get:

D ≈ ps(0, ϕ) πa^2 / (2πR dz)

D ≈ (ps(0, ϕ) a^2) / (2R dz)

Since we are interested in the value of D on the z-axis (R = ∞), we have:

D ≈ (ps(0, ϕ) a^2) / (2(∞) dz)

D ≈ (ps(0, ϕ) a^2) / (∞)

As the denominator approaches infinity, we can approximate D as:

D ≈ (ps(0, ϕ) a^2) / ∞ = 0

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The impedance of the circuit is 336.2 ohms. The rms current through the resistor is 0.342 A. The average power dissipated in the circuit is 39.2 W. The peak current through the resistor is 0.484 A. The peak voltage across the inductor is 68.7 V. The peak voltage across the capacitor is 19.6 V. The new resonance frequency is 60.0 Hz.

To find the impedance of the circuit, we need to consider the combined effects of the inductor, capacitor, and resistor. The impedance of an RL circuit is given by Z = [tex]\sqrt{(R^2 + (ωL - 1/(ωC))^2)}[/tex], where R is the resistance, ω is the angular frequency (2πf), L is the inductance, and C is the capacitance. Plugging in the values, we get Z = [tex]\sqrt{(336^2 + (2\pi (60)(0.200) - 1/(2\pi (60)(4.60 x 10^-6)))^2)}[/tex] ≈ 336.2 ohms.

The rms current through the resistor can be calculated using Ohm's law, where I = V/Z, with V being the rms voltage supplied by the generator. So, I = 115 V / 336.2 ohms ≈ 0.342 A.

The average power dissipated in the circuit can be determined using the formula P = I^2R, where P is power and R is the resistance. Thus, P = [tex](0.342 A)^2[/tex] x 336.2 ohms ≈ 39.2 W.

The peak current through the resistor is equal to the rms current multiplied by the square root of 2. Therefore, the peak current is approximately 0.342 A x [tex]\sqrt{2}[/tex] ≈ 0.484 A.

The peak voltage across an inductor is given by V_L = I_LωL, where I_L is the peak current through the inductor. Since the inductor is in series with the resistor, the peak current is the same as the peak current through the resistor. Thus, V_L = 0.484 A x 2π(60)(0.200 H) ≈ 68.7 V.

The peak voltage across a capacitor is given by V_C = I_C/(ωC), where I_C is the peak current through the capacitor. Again, since the capacitor is in series with the resistor, the peak current is the same as the peak current through the resistor. Therefore, V_C = 0.484 A / (2π(60)(4.60 x 10^-6 F)) ≈ 19.6 V.

When the circuit is in resonance, the reactances of the inductor and capacitor cancel each other out, resulting in a purely resistive impedance. At resonance, the angular frequency ω is given by ω = 1/sqrt(LC). Plugging in the values of L and C, we find ω = 1/[tex]\sqrt{0.200 H x 4.60 x 10^-6 F }[/tex]≈ 60.0 Hz, which is the new resonance frequency4

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The apparent power (kVA) is decreased as a result, while the active power (kW) that is available for practical work is increased. The phase advancer decreases the reactive power needed by producing more magnetizing flux. The motor's power factor improves as a result.

The most frequent method for enhancing the power factor of an AC electrical system is the employment of a capacitor bank. In the circuit schematic, the inductive load is connected in parallel with capacitors, usually at the consumption point. Here is a short description of how it works:

(1)Certain components or loads (such as motors and transformers) in an AC electrical system have inductive properties that create a phase shift in the relationship between voltage and current. A trailing power factor, which is caused by this phase shift, can be wasteful and raise energy expenses.

The reactive power supplied by the capacitors helps balance the reactive power required by the inductive load when a capacitor bank is connected in parallel with the load. By doing this, the phase shift is balanced and the power factor is raised to a value closer to unity (1.0).

Capacitors provide leading reactive current, which balances out the inductive load's trailing reactive current. The apparent power (kVA) is decreased as a result, while the active power (kW) that is available for practical work is increased.

(2)Enhancing Power Factor using Synchronous Condenser:

A revolving device called a synchronous condenser, often referred to as a synchronous compensator, aids in raising an electrical system's power factor. Here is a quick rundown of how it functions:

In essence, a synchronous condenser is a synchronous motor that doesn't require a mechanical load to run. It is made up of a field winding that is stimulated by a DC power source and a rotor that is linked to the power system.

A synchronous condenser is introduced to a system and over-excited by raising the field current when the power factor of the system is behind. Reactive power is produced by the synchronous condenser as a result.

The system's trailing reactive power is made up for by the reactive power generated by the synchronous condenser, which significantly raises the power factor.

The synchronous condenser may alter the amount of provided reactive power by adjusting the field excitation, providing fine control over the power factor.

(3)Power Factor Improvement using Phase Advancers:

Phase advancers are typically used in induction motors to improve their power factor during starting and low-load conditions. Here's a simplified explanation:

A phase advancer is a tool that adds more magnetizing flux to an induction motor's rotor circuit during startup or low-load operation.

A capacitor and an auxiliary winding coupled in line with the motor's primary winding make up the phase advancer.

Phase shifting occurs between the currents in the main and auxiliary windings when the capacitor is connected to the auxiliary winding during starting. A spinning magnetic field is created by this phase shift, which helps to generate the initial torque.

The phase advancer decreases the reactive power needed from the power supply by producing more magnetizing flux. The motor's power factor improves as a result.

These are the basic principles of power factor improvement using capacitor banks, synchronous condensers, and phase advancers.

The circuit diagram is given in image.

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A 0.150 kg cube of ice (frozen water) is floating in glycerin. The glycerin is in a tall cylinder that has inside radius 3.50 cm. The level of the glycerin is well below the top of the cylinder. The change in height of the liquid in the cylinder when the ice completely melts is approximately 0.129 meters.

Let's calculate the change in height of the liquid in the cylinder when the ice cube completely melts.

Given:

Mass of the ice cube (m) = 0.150 kg

Radius of the cylinder (r) = 3.50 cm = 0.035 m

To calculate the change in height, we need to determine the volume of the ice cube. Since the ice is floating, its volume is equal to the volume of the liquid it displaces.

Density of water (ρ_water) = 1000 kg/m^3 (approximately)

Volume of the ice cube (V_ice) = m / ρ_water

V_ice = 0.150 kg / 1000 kg/m^3 = 0.000150 m^3

Next, we can calculate the change in height of the liquid in the cylinder when the ice melts.

Change in height (Δh) = V_ice / (π × r^2)

Δh = 0.000150 m^3 / (π × (0.035 m)^2)

Δh ≈ 0.129 m

The change in height of the liquid in the cylinder when the ice completely melts is approximately 0.129 meters.

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a)the magnitude of the magnetic field.B = 3.53 x 10^(-3) T and the magnetic field is directed in the negative z-direction.b)Work done by the magnetic field is zero because the magnetic field is perpendicular to the direction of motion.c) the time taken.t = 7.43 x 10^(-8) s.

A magnetic field that will cause the electron to cross x = 42 cm is given by (a) and (b). What work is done on the electron during this motion and how long will the trip take from the y-axis to the x-axis? Find the magnitude and direction of the magnetic field.Answer:Magnitude of magnetic field = 3.53 x 10^(-3) TDirection of magnetic field = Inverted in z-direction.

Work done = 0JTime taken = 7.43 x 10^(-8) sStep-by-step

A force exists on a charged particle due to the magnetic field, which results in circular motion. The strength of the magnetic force is given by the equation Fm = qvBsinθ, where q is the charge on the particle, v is the velocity of the particle, B is the magnetic field strength, and θ is the angle between the velocity vector and the magnetic field vector.

Lorentz force is the result of the magnetic force acting on a charged particle in a magnetic field, which causes the particle to move in a circle, as shown below:Fm = q(v×B)Here, B is the magnetic field vector, which is perpendicular to the plane of the paper. As a result, the force on the particle is perpendicular to its velocity vector and is directed towards the center of the circle.Force = maSo, ma = q(v×B)From this we get acceleration of the charged particle due to magnetic field.

By using this acceleration we can calculate the radius of the circle that the electron moves. As the path of electron is circular, centripetal force must be equal to the magnetic force.Fc = FmBy using these we can calculate the magnetic field magnitude, direction and work done and time taken.

(a) Magnitude and direction of the magnetic fieldAs the magnetic force is the centripetal force we haveFc = FmFrom this we getqvB = mv^2 / rB = mv / qr = mv / qBvSubstitute the values givenm = 9.11 x 10^(-31)kgq = 1.60 x 10^(-19) C x = 42 cm = 0.42 mT = 2.35 x 10^(-6) sB = m * v / (q * r)Calculate the magnitude of the magnetic field.B = 3.53 x 10^(-3) T

We know that the force is perpendicular to the velocity and the direction of the magnetic field is given by the right-hand rule. In the z-direction, the velocity vector is towards the observer, and the magnetic force vector is in the opposite direction to the observer. As a result, the magnetic field is directed in the negative z-direction.

(b) Work done by the magnetic field is zero because the magnetic field is perpendicular to the direction of motion. The magnetic field only causes a change in direction.

(c) As the magnetic force is the centripetal force we haveqvB = mv^2 / rBy substituting the valuesq = 1.60 x 10^(-19) Cv = 3.0 x 10^6 m/sm = 9.11 x 10^(-31) kgB = 3.53 x 10^(-3) Tr = 0.42 m Calculate the time taken.t = 7.43 x 10^(-8) s

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The potential difference between the points (10cm, 5.0cm) and (5.0cm, 5.0cm) due to the point charge Q=20 nC at the origin is 400 V.

To calculate the potential difference between the given points, we can use the formula for the electric potential due to a point charge. The formula states that the potential difference (V) between two points is equal to the charge (Q) divided by the distance (r) between the points. In this case, the charge Q is 20 nC and the distance between the points is 5.0cm.

First, we need to calculate the distance between the two points. The points lie on the same horizontal line, so the distance between them is simply the difference in their x-coordinates. The distance is (10cm - 5.0cm) = 5.0cm.

Next, we substitute the values into the formula. The potential difference (V) is equal to (20 nC) divided by (5.0cm). Remember to convert the distance to meters, as the SI unit for charge is coulombs. 1 cm = 0.01 m, so 5.0cm = 0.05m.

Calculating the potential difference, V = (20 nC) / (0.05m) = 400 V.

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A 1.0nF air-filled parallel plate capacitor is charged up by a 100V battery. While still connected to the battery, a dielectric with κ=3 is filled between the plates. Therefore, the correct option is A. 15 μJ.

The capacitance of an air-filled parallel plate capacitor is given by the formula C = εA/d,

where ε is the permittivity of air, A is the area of the plates, and d is the distance between them.

The permittivity of air is 8.85 x 10^-12 F/m.So,C = εA/d = 8.85 x 10^-12 * A/d = 1.0 x 10^-9nF = 1 x 10^-12 F So, A/d = 1.13 x 10^-3 m^-1 = capacitance per meter.

Since the capacitor is charged to 100V, the energy stored in it is given by the formulaE = 1/2 * CV^2 = 1/2 * 1 x 10^-12 * (100)^2 = 5 x 10^-9 J.

When a dielectric material with a dielectric constant (κ) is introduced between the plates, the capacitance of the capacitor increases by a factor of κ, which means the capacitance of the capacitor becomes κC, and the final energy stored in the capacitor is E' = 1/2 * κCV^2 = 1/2 * 3 * 1 x 10^-12 * (100)^2 = 1.5 x 10^-8 J = 15 μJ.

Therefore, the correct option is A. 15 μJ.

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Power transmitted by the steel propeller shaft = 5.5 MW = 5.5 x 10^6 W.

Speed of rotation = 180 rpm.

Shearing stress = 60 MPa.

Maximum angle of twist = 1°

Length of the steel propeller shaft = 25 diameters.

Given that modulus of rigidity of the steel propeller shaft G = 83 GPa.

We know that the power transmitted by the shaft, P = 2πNT/60,where N = speed of rotation in rpm and T = torque in N-m.

Substituting the given values, we get,5.5 x 10^6 = 2π x 180T/60.

T = 2.05 x 10^7 N-m. Now, we know that the maximum shearing stress τmax = 16T/πd^3 and maximum angle of twist θmax = TL/Gd^4.

Now, substituting the given values : we get,τmax = 16T/πd^3 = 60 MPa.θmax = TL/Gd^4 = 1° x π/180 x 25d = 25d.

Solving for diameter d, we get, τmax = 16T/πd^3⇒ 60 x 10^6 = 16 x 2.05 x 10^7/πd^3

⇒ d^3 = 2.69 x 10^-3

⇒ d = 0.144 m or 144 mm.

Answer: Diameter of the steel propeller shaft = 144 mm.

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The maximum speed of each mass when it has moved free of the spring is approximately 0.431 m/s.

To find the maximum speed of each mass when it has moved free of the spring, we can use the principle of conservation of mechanical energy.

When the masses are pressed against the spring, the potential energy stored in the spring is given by the equation:

PE = (1/2)kx^2

Where PE is the potential energy, k is the force constant of the spring, and x is the compression or extension of the spring from its normal length.

In this case, the compression of the spring is 15.0 cm, or 0.15 m. The force constant is given as 1.65 N/cm, or 16.5 N/m. So the potential energy stored in the spring is:

PE = (1/2)(16.5 N/m)(0.15 m)^2 = 0.1485 J

According to the conservation of mechanical energy, this potential energy is converted into the kinetic energy of the masses when they are free of the spring.

The kinetic energy of an object is given by the equation:

KE = (1/2)mv^

Where KE is the kinetic energy, m is the mass, and v is the velocity of the object.

Since the masses are identical, each mass will have the same kinetic energy and maximum speed.

Setting the potential energy equal to the kinetic energy:

0.1485 J = (1/2)(1.60 kg)v^2

Solving for v:

v^2 = (2 * 0.1485 J) / (1.60 kg)

v^2 = 0.185625 J/kg

v = √(0.185625 J/kg) ≈ 0.431 m/s

Therefore, the maximum speed of each mass when it has moved free of the spring is approximately 0.431 m/s.

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when a negatively charged particle is free, it will move in the direction of the electric field, which is towards regions of the opposite charge.

When free, a negative charge tries to move from high potential to low potential, as it is attracted towards a region of opposite charge. This is known as the direction of the electric field.A negatively charged particle can move in a range of directions. When it is free to move, it will move in a direction that brings it to a position of lower potential energy. This is due to the fact that electric potential energy is inversely related to electric potential. Electric potential is the energy that a charged particle has as a result of its location in an electric field. When a particle is in an electric field, it will experience a force that pushes it in the direction of the region of opposite charge. The direction of the electric field is defined as the direction that a positively charged particle would move if it were free to do so.The particle would be attracted to regions of the opposite charge and repelled from regions of the same charge. Therefore, when a negatively charged particle is free, it will move in the direction of the electric field, which is towards regions of the opposite charge.

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The magnetic field inside a solenoid of length 3.00 cm and radius 0.950 cm with 49 turns and a wire that has 1.35 amps of current is 0.449 T.

The magnetic energy density inside the solenoid is 0.180 J/m³.

The magnetic field inside a solenoid can be given as; B = μ₀*n*I, Where;

B is the magnetic field, n is the number of turns per unit length, I is the currentμ₀ is the magnetic constant or permeability of free space.

We know that the length of the solenoid l = 3.00 cm and radius r = 0.950 cm, thus we can calculate the number of turns per unit length, n = N/l = 49/0.03 = 1633.33 turns/m

We know the current I is 1.35 ampsNow, using the formula,

B = μ₀*n*I

We can substitute the given values to obtain;

B = μ₀*n*I= 4π × 10⁻⁷ T*m/A × 1633.33 turns/m × 1.35

A= 0.449 T

Therefore, the magnitude of the magnetic field inside the solenoid is 0.449 T.

The magnetic energy density inside a magnetic field can be given as;u = (B²/2μ₀)We know the magnetic field inside the solenoid is 0.449 T, substituting this and the value of μ₀ = 4π × 10⁻⁷ T*m/A, we get;u = (B²/2μ₀) = (0.449²/2 × 4π × 10⁻⁷) = 0.180 J/m³

Therefore, the magnetic energy density inside the solenoid is 0.180 J/m³.

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Sound waves entering human ear first pass through the auditory canal before reaching the eardrum. If a typical adult has an auditory canal of 2.5cm long and 7.0mm in diameter, suppose that when you listen to ordinary conversation, the intensity of sound waves is about 3.2 × 10−6W/m^2 ,, the average power delivered to the eardrum when listening to ordinary conversation is approximately 1.23 × 10^(-10) Watts.

To calculate the average power delivered to the eardrum, we can use the formula:

Power = Intensity× Area

Given:

Intensity (I) = 3.2 × 10^(-6) W/m^2

Auditory canal length (L) = 2.5 cm = 0.025 m

Auditory canal diameter (d) = 7.0 mm = 0.007 m

First, we need to find the area of the cross-section of the auditory canal. Since the canal has a circular cross-section, the area can be calculated using the formula:

Area = π × (d/2)^2

Substituting the given values:

Area = π × (0.007/2)^2

Area ≈ 3.85 × 10^(-5) m^2

Now we can calculate the power delivered to the eardrum:

Power = Intensity × Area

Power = (3.2 × 10^(-6) W/m^2) * (3.85 × 10^(-5) m^2)

Power ≈ 1.23 × 10^(-10) W

Therefore, the average power delivered to the eardrum when listening to ordinary conversation is approximately 1.23 × 10^(-10) Watts.

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0.25 -uF parallel plate capacitor is connected to a 120 V battery.  the charge on one of the capacitor plates is 30 μC.

To find the charge on one of the capacitor plates, we can use the equation Q = CV, where Q represents the charge, C is the capacitance, and V is the voltage.

Given that the capacitance is 0.25 μF (microfarads) and the voltage is 120 V, we can substitute these values into the equation to find the charge:

Q = (0.25 μF) * (120 V)

  = 30 μC (microcoulombs)

Therefore, the charge on one of the capacitor plates is 30 μC.

To explain this further, a capacitor stores electrical charge when a voltage is applied across its plates. The capacitance (C) of a capacitor is a measure of its ability to store charge. In this case, the given capacitance is 0.25 μF.

When the capacitor is connected to a 120 V battery, the voltage across the capacitor plates is 120 V. By multiplying the capacitance by the voltage, we obtain the charge stored on one of the plates, which is 30 μC.

This means that the capacitor is capable of storing 30 microcoulombs of charge when connected to a 120 V battery. The charge remains on the plates until the capacitor is discharged or the voltage across the plates is changed.

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Answer: The altitude is 3.29 × 106 m below the surface of Earth.

The radius of the Earth RE=6.378×10⁶m

acceleration due to gravity at its surface is 9.81 m/s². The expression that relates the acceleration due to gravity with the distance from the center of Earth is given by:

g = (GM)/r²

Where g is the acceleration due to gravity, G is the universal gravitational constant (6.67 × 10-11 Nm²/kg²), M is the mass of Earth, and r is the distance from the center of Earth.

We can solve for r to find the distance from the center of Earth at which the acceleration due to gravity is 2.6 m/s²:

g = (GM)/r²r²

= GM/g

Let's plug in the given values to solve for r:

r² = (6.67 × 10-11 Nm²/kg² × 5.97 × 1024 kg)/(2.6 m/s²)

r² = 9.56 × 1012 m²

r = 3.09 × 106 m.

Now we can find the altitude above the surface of Earth by subtracting the radius of Earth from r:

Altitude = r - RE

Altitude = 3.09 × 106 m - 6.378 × 106 m.

Altitude = -3.29 × 106 m.

This is a negative value, which means that the acceleration due to gravity of 2.6 m/s² is found at a distance below the surface of Earth.

So, the altitude is 3.29 × 106 m below the surface of Earth.

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The total flux leaving the cube is 8.4×10⁴ Nm²/C.

a. To draw point charges and cube at their respective locations, the following plot can be used:

Image plot of point charges and cube.

b. The total flux leaving the cube is to be determined. The flux leaving the cube due to each charge will be calculated first. Total flux will be the algebraic sum of the flux due to all three charges. Mathematically, it is given by:

ϕ = ϕ1 + ϕ2 + ϕ3

The electric flux due to a point charge is given by:

ϕ = q / (ε₀ * r²)

Where q is the charge of the point charge, ε₀ is the permittivity of free space, and r is the distance between the point charge and the cube.

Therefore, using the above equation, the electric flux due to each point charge can be calculated as:

q₁ = 15 nC, r₁ = √(1 + 1.25² + 0.5²) = 1.68 m

q₂ = 12 nC, r₂ = √(2.25² + 1² + 1.25²) = 2.76 m

q₃ = -12 nC, r₃ = √(1² + 0.5² + 1.25²) = 1.62 m

Substituting the values in the above equation,

ϕ₁ = (15×10⁻⁹) / (8.854×10⁻¹² * 1.68²) = 2.08×10⁶ Nm²/C

ϕ₂ = (12×10⁻⁹) / (8.854×10⁻¹² * 2.76²) = 1.05×10⁶ Nm²/C

ϕ₃ = (-12×10⁻⁹) / (8.854×10⁻¹² * 1.62²) = -2.29×10⁶ Nm²/C

Total Flux ϕ = ϕ₁ + ϕ₂ + ϕ₃

ϕ = 2.08×10⁶ + 1.05×10⁶ - 2.29×10⁶ = 8.4×10⁴ Nm²/C

Thus, the total flux leaving the cube is 8.4×10⁴ Nm²/C.

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