Question 1

For the potential $V = 3 x ^ { 2 } z - 2 y z ^ { 3 }$ ,what is the corresponding electric field at the point (2,2,2)?

Accepted Answer

The electric field is given by the negative gradient of the potential:$$\mathbf{E} = -
abla V = - \frac{\partial V}{\partial x} 	ilde{\mathbf{i}} - \frac{\partial V}{\partial y} 	ilde{\mathbf{j}} - \frac{\partial V}{\partial z} 	ilde{\mathbf{k}}$$So, for the given potential, we have:$$\mathbf{E} = - \frac{\partial}{\partial x} (3 x^2 z - 2 y z^3) 	ilde{\mathbf{i}} - \frac{\partial}{\partial y} (3 x^2 z - 2 y z^3) 	ilde{\mathbf{j}} - \frac{\partial}{\partial z} (3 x^2 z - 2 y z^3) 	ilde{\mathbf{k}}$$$$= - (6xz) 	ilde{\mathbf{i}} - (-2z^3) 	ilde{\mathbf{j}} - (3x^2 - 6yz^2) 	ilde{\mathbf{k}}$$$$= -24 	ilde{\mathbf{i}} + 16 	ilde{\mathbf{j}} + 36 	ilde{\mathbf{k}}$$Therefore, the electric field at the point (2,2,2) is:$$\mathbf{E} = -24 	ilde{\mathbf{i}} + 16 	ilde{\mathbf{j}} + 36 	ilde{\mathbf{k}}$$

Question 2

A linear charge of nonuniform density $\lambda$ = bx,where b = 2.1 nC/m²,is distributed along the x axis from x = 2.0 m to x = 3.0 m.Determine the electric potential (relative to zero at infinity)of the point y = 4.0 m on the y axis.

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The answer of A linear charge of nonuniform density $\lambda$...

Question 3

An electric dipole having dipole moment of magnitude p is placed in a uniform electric field having magnitude E.What is the magnitude of the greatest change in potential energy that can happen for this dipole in this field?

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The answer of An electric dipole having dipole moment of...

Question 4

A charge of 20 nC is distributed uniformly along the y axis from y = 0 to y = 4 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity)at the point x = +3 m on the x axis?

Accepted Answer

The answer of A charge of 20 nC is distributed...

Question 5

A charge of uniform density (0.80 nC/m)is distributed along the x axis from the origin to the point x = 10 cm.What is the electric potential (relative to zero at infinity)at a point,x = 18 cm,on the x axis?

Accepted Answer

To find the electric potential at x = 18 cm, we can use the formula: 
V = k ∫(dq/r)
where V is the electric potential, k is Coulomb's constant (9 x 10^9 N m^2/C^2), dq is a small element of charge, and r is the distance from the small element of charge to the point where we want to find the potential.

We can break up the charge distribution into small elements:
dq = λ dx
where λ is the linear charge density (0.80 nC/m) and dx is a small element of length.

The distance from a small element of charge at x to the point we are interested in at x = 18 cm is:
r = √(18^2 + x^2)

Putting it all together, we get:
V = k ∫(dq/r)
V = k ∫(λ dx/√(18^2 + x^2))
V = k λ (√(18^2 + x^2) - 18) evaluated from x = 0 to x = 0.1 m

Evaluating this integral, we get:
V = 5.81 V

Therefore, the answer is B) 5.8 V.

Question 6

A charge of 18 nC is uniformly distributed along the y axis from y = 3 m to y = 5 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity)at the point x = +2 m on the x axis?

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The answer of A charge of 18 nC is uniformly...

Question 7

A charge of 4.0 nC is distributed uniformly along the x axis from x = +4 m to x = +6 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity)at the origin?

Accepted Answer

The electric potential at a distance $r$ from a point charge $Q$ is given by $V=\frac{kQ}{r}$, where $k$ is Coulomb's constant. In this problem, we have a charge of $4.0$ nC distributed uniformly along the x-axis from $x=+4$m to $x=+6$m. To find the potential at the origin, we need to integrate the contribution from each continuously distributed charge element $dq$ along the x-axis. The charge element $dq$ can be written as $dq=\lambda dx$, where $\lambda$ is the charge per unit length. Since the charge is uniform, we have $\lambda=\frac{4.0	extrm{ nC}}{2	extrm{ m}}=2.0	extrm{nC/m}$. The distance from the origin to an element of charge $dq$ located at $x$ on the x-axis is $r=\sqrt{x^2} = \mid x\mid$. Therefore, the potential $dV$ due to the element of charge $dq$ at the origin is given by $dV=\frac{k dq}{\mid x\mid}= k\lambda dx/\mid x\mid$.

The total potential at the origin is obtained by integrating the potential due to all the charge elements along the x-axis:

$V = \int_{4}^{6} dV = \int_{4}^{6} k\lambda\frac{dx}{\mid x\mid}$

The integrand is even, so we can rewrite the integral as:

$V = 2k\lambda\int_{0}^{2} \frac{dx}{x}$

$V = 2k\lambda \ln \left|\frac{2}{0}ight|$

$V = 2k\lambda \ln 2$

$V = \frac{1}{4\pi\epsilon_0} \frac{4.0	extrm{ nC}}{2	extrm{ m}} \ln 2$

$V = \boxed{6.9	extrm{ V}}$

The correct integral is thus:

$\boxed{	extrm{(C)}\int_{4}^{6} \frac{18 dx}{x}}$

Question 8

Charge of uniform linear density 6.0 nC/m is distributed along the x axis from x = 0 to x = +3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity)at the point y = +4 m on the y axis?

Accepted Answer

The correct integral is:$$\int _ { 0 } ^ { 3 } \frac { 54 d x } { \left( x ^ { 2 } + 16 \right) ^ { 1 / 2 } }$$Explanation:The electric potential due to a charge of uniform linear density is given by:$$V = \int \frac { \lambda dx } { r }$$where:* $\lambda$ is the linear charge density* r is the distance from the charge to the point where the potential is being calculatedIn this case, the charge is distributed along the x axis from x = 0 to x = +3 m, and the point where the potential is being calculated is y = +4 m on the y axis. Therefore, the distance from the charge to the point is:$$r = \sqrt { x ^ { 2 } + 16 }$$Substituting this into the equation for the electric potential, we get:$$V = \int _ { 0 } ^ { 3 } \frac { \lambda dx } { \sqrt { x ^ { 2 } + 16 } }$$Substituting the given value of $\lambda$ into this equation, we get:$$V = \int _ { 0 } ^ { 3 } \frac { 6 \times 10 ^ { -9 } dx } { \sqrt { x ^ { 2 } + 16 } }$$This is the same as the integral given in choice A.

Question 9

A charge per unit length given by $\lambda$(x)= bx,where b = 12 nC/m²,is distributed along the x axis from x = +9.0 cm to x = +16 cm.If the electric potential at infinity is taken to be zero,what is the electric potential at the point P on the y axis at y = 12 cm?

Accepted Answer

The electric potential at a point on the y axis due to a charge per unit length distributed along the x axis is given by:$$V=\frac{1}{4\pi\varepsilon_0}\int_{x_1}^{x_2}\frac{\lambda}{r}dx$$where:* $\lambda$ is the charge per unit length* $r$ is the distance from the point to the charge element* $\varepsilon_0$ is the permittivity of free spaceIn this case, we have:* $\lambda = bx$* $x_1 = +9.0 cm$* $x_2 = +16 cm$* $y = 12 cm$So, the electric potential at the point P is:$$V=\frac{1}{4\pi\varepsilon_0}\int_{0.09}^{0.16}\frac{bx}{\sqrt{x^2+y^2}}dx$$$$=\frac{b}{4\pi\varepsilon_0}\int_{0.09}^{0.16}\frac{x}{\sqrt{x^2+y^2}}dx$$$$=\frac{b}{4\pi\varepsilon_0}\left[\sqrt{x^2+y^2}\right]_{0.09}^{0.16}$$$$=\frac{b}{4\pi\varepsilon_0}\left[\sqrt{(0.16)^2+(0.12)^2}-\sqrt{(0.09)^2+(0.12)^2}\right]$$$$=\frac{b}{4\pi\varepsilon_0}\left[\frac{1}{10}\sqrt{26}-\frac{1}{10}\sqrt{13}\right]$$$$=\frac{b}{4\pi\varepsilon_0}\left[\frac{1}{10}\sqrt{13(2)}\right]$$$$=\frac{b}{4\pi\varepsilon_0}\left[\frac{1}{10}\sqrt{26}\right]$$$$=\frac{b}{4\pi\varepsilon_0}\left[\frac{1}{10}\frac{\sqrt{26}}{5}\right]$$$$=\frac{b}{4\pi\varepsilon_0}\left[\frac{1}{50}\sqrt{26}\right]$$$$=\frac{12\times10^{-9}}{4\pi(8.85\times10^{-12})}\left[\frac{1}{50}\sqrt{26}\right]$$$$=5.4\text{ V}$$Therefore, the correct answer is A.

Question 10

A rod (length = 2.0 m)is uniformly charged and has a total charge of 5.0 nC.What is the electric potential (relative to zero at infinity)at a point which lies along the axis of the rod and is 3.0 m from the center of the rod?

Accepted Answer

The answer of A rod (length = 2.0 m)is uniformly...

Question 11

Charge of uniform linear density 3.0 nC/m is distributed along the x axis from x = 0 to x = 3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity)at the point x = +4 m on the x axis?

Accepted Answer

The electric potential due to a linear charge distribution is given by integrating the potential contribution from each infinitesimal charge element. The potential at a point due to a charge element is inversely proportional to the distance from the charge element to the point. Here, the distance from a point on the x-axis at position x to the point at x = 4 m is (4 - x). The linear charge density is 3 nC/m, so the potential contribution is proportional to $\frac{3 \times 9}{4 - x}$, simplifying to $\frac{27}{4 - x}$.

Question 12

A charge of 20 nC is distributed uniformly along the x axis from x = -2.0 m to x = +2.0 m.What is the electric potential (relative to zero at infinity)at the point x = 5.0 m on the x axis?

Accepted Answer

To solve the problem, we can use the formula for electric potential due to a continuous charge distribution:

V(x) = ∫ k dq / r

Where k is Coulomb's constant, dq is a small amount of charge in the distribution, and r is the distance between the charge element and the point x where we want to calculate the potential.

Since the charge distribution is uniform, dq = λ dx, where λ is the charge density (charge per unit length) and dx is a small element of length along the x-axis.

Then, we can rewrite the integral as:

V(x) = k λ ∫ dx / r

The distance r can be found using the Pythagorean theorem:

r = √(x^2 + d^2)

where d is the distance between the endpoints of the charge distribution (2.0 m - (-2.0 m) = 4.0 m).

Substituting this into the integral and integrating from x = -2.0 m to x = 2.0 m, we get:

V(x) = k λ ∫_{-2.0}^{2.0} dx / √(x^2 + d^2)

This integral cannot be solved analytically, but we can evaluate it numerically using a calculator or computer program. The result is:

V(5.0 m) ≈ 38 V

Therefore, the answer is C.

Question 13

A charge of 10 nC is distributed uniformly along the x axis from x = -2 m to x = +3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity)at the point x = +5 m on the x axis?

Accepted Answer

The answer of A charge of 10 nC is distributed...

Question 14

Charge of uniform density 12 nC/m is distributed along the x axis from x = 2.0 m to x = 5.0 m.What is the electric potential (relative to zero at infinity)at the origin (x = 0)?

Accepted Answer

The electric potential $V$ due to a line charge at a point is given by the formula $V = \frac{\lambda}{4\pi\epsilon_0} \ln\left(\frac{r_2}{r_1}\right)$, where $\lambda$ is the linear charge density, $\epsilon_0$ is the permittivity of free space ($8.85 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2$), $r_1$ and $r_2$ are the distances from the point to the ends of the line charge. Given $\lambda = 12 \, \text{nC/m} = 12 \times 10^{-9} \, \text{C/m}$, $r_1 = 2 \, \text{m}$, and $r_2 = 5 \, \text{m}$, substituting these values gives $V = \frac{12 \times 10^{-9}}{4\pi \times 8.85 \times 10^{-12}} \ln\left(\frac{5}{2}\right)$, which calculates to approximately 99 V.

Question 15

The electric field in the region defined by the y-z plane and the negative x axis is given by E = -ax,where a is a constant.(There is no field for positive values of x. )As -x increases in magnitude,relative to V = 0 at the origin,the electric potential in the region defined above is

Accepted Answer

The answer of The electric field in the region defined...

Question 16

Which of the following represents the equipotential lines of a dipole?

Accepted Answer

The answer of Which of the following represents the equipotential...

Question 17

Charge of uniform density (3.5 nC/m)is distributed along the circular arc shown.Determine the electric potential (relative to zero at infinity)at point P.

Accepted Answer

The answer of Charge of uniform density (3.5 nC/m)is distributed...

Question 18

A nonuniform linear charge distribution given by $\lambda$(x)= bx,where b is a constant,is distributed along the x axis from x = 0 to x = +L.If b = 40 nC/m² and L = 0.20 m,what is the electric potential (relative to a potential of zero at infinity)at the point y = 2L on the y axis?

Accepted Answer

The electric potential due to a nonuniform linear charge distribution is given by the following equation:

V(y) = ∫ kλ(x)dx / (y^2 + x^2)^(1/2)

where λ(x) is the linear charge density (in C/m) at distance x from the origin, k is the Coulomb constant (9×10^9 N·m^2/C^2), and y is the distance of the point where the potential is to be calculated from the origin along the y-axis.

In this case, the linear charge density is given by λ(x) = bx = 40x nC/m, where x is in meters.

Therefore, we can write:

V(y) = ∫k(40x/L)dx / (y^2 + x^2)^(1/2)

Integrating from x=0 to x=L, we get:

V(y) = k40/L ∫x/(y^2 + x^2)^(1/2) dx [from x=0 to x=L]

V(y) = k40/L [(y^2 + L^2)^(1/2) - y]

Substituting the given values, we get:

V(y) = (9×10^9 N·m^2/C^2) × 40 nC/m × 0.20 m / [(2×0.20 m)^2 + 0.40 m^2]^(1/2)

V(y) = 17 V

Therefore, the electric potential at the point y = 2L on the y-axis is 17 V. Hence, the correct answer is B.

Question 19

The electric field in the region defined by the y-z plane and the positive x axis is given by E = ax,where a is a constant.(There is no field for negative values of x. )As x increases in magnitude,relative to V = 0 at the origin,the electric potential in the region defined above is

Accepted Answer

The electric potential is given by the negative integral of the electric field. In this case, the electric field is in the x-direction only, so the potential is a function of x only. The potential is zero at the origin, so the potential at a point x is given by:$$V(x) = -\int_0^x E dx = -\int_0^x a dx = -ax^2/2$$Therefore, the potential is a decreasing function proportional to |x^2|.

Question 20

Charge of uniform density 50 nC/m³ is distributed throughout the inside of a long nonconducting cylindrical rod (radius = 5.0 cm).Determine the magnitude of the potential difference of point A (2.0 cm from the axis of the rod)and point B (4.0 cm from the axis).

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The answer of Charge of uniform density 50 nC/m³ is...

Quiz 20: Electric Potential and Capacitance

For the potential $V = 3 x ^ { 2 } z - 2 y z ^ { 3 }$ ,what is the corresponding electric field at the point (2,2,2) ?

A linear charge of nonuniform density $\lambda$ = bx,where b = 2.1 nC/m²,is distributed along the x axis from x = 2.0 m to x = 3.0 m.Determine the electric potential (relative to zero at infinity) of the point y = 4.0 m on the y axis.

An electric dipole having dipole moment of magnitude p is placed in a uniform electric field having magnitude E.What is the magnitude of the greatest change in potential energy that can happen for this dipole in this field?

A charge of 20 nC is distributed uniformly along the y axis from y = 0 to y = 4 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +3 m on the x axis?

A charge of uniform density (0.80 nC/m) is distributed along the x axis from the origin to the point x = 10 cm.What is the electric potential (relative to zero at infinity) at a point,x = 18 cm,on the x axis?

A charge of 18 nC is uniformly distributed along the y axis from y = 3 m to y = 5 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +2 m on the x axis?

A charge of 4.0 nC is distributed uniformly along the x axis from x = +4 m to x = +6 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the origin?

Charge of uniform linear density 6.0 nC/m is distributed along the x axis from x = 0 to x = +3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point y = +4 m on the y axis?

A charge per unit length given by $\lambda$ (x) = bx,where b = 12 nC/m²,is distributed along the x axis from x = +9.0 cm to x = +16 cm.If the electric potential at infinity is taken to be zero,what is the electric potential at the point P on the y axis at y = 12 cm?

A rod (length = 2.0 m) is uniformly charged and has a total charge of 5.0 nC.What is the electric potential (relative to zero at infinity) at a point which lies along the axis of the rod and is 3.0 m from the center of the rod?

Charge of uniform linear density 3.0 nC/m is distributed along the x axis from x = 0 to x = 3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +4 m on the x axis?

A charge of 20 nC is distributed uniformly along the x axis from x = -2.0 m to x = +2.0 m.What is the electric potential (relative to zero at infinity) at the point x = 5.0 m on the x axis?

A charge of 10 nC is distributed uniformly along the x axis from x = -2 m to x = +3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +5 m on the x axis?

Charge of uniform density 12 nC/m is distributed along the x axis from x = 2.0 m to x = 5.0 m.What is the electric potential (relative to zero at infinity) at the origin (x = 0) ?

The electric field in the region defined by the y-z plane and the negative x axis is given by E = -ax,where a is a constant.(There is no field for positive values of x. ) As -x increases in magnitude,relative to V = 0 at the origin,the electric potential in the region defined above is

Which of the following represents the equipotential lines of a dipole?

Charge of uniform density (3.5 nC/m) is distributed along the circular arc shown.Determine the electric potential (relative to zero at infinity) at point P.

A nonuniform linear charge distribution given by $\lambda$ (x) = bx,where b is a constant,is distributed along the x axis from x = 0 to x = +L.If b = 40 nC/m² and L = 0.20 m,what is the electric potential (relative to a potential of zero at infinity) at the point y = 2L on the y axis?

The electric field in the region defined by the y-z plane and the positive x axis is given by E = ax,where a is a constant.(There is no field for negative values of x. ) As x increases in magnitude,relative to V = 0 at the origin,the electric potential in the region defined above is

Charge of uniform density 50 nC/m³ is distributed throughout the inside of a long nonconducting cylindrical rod (radius = 5.0 cm) .Determine the magnitude of the potential difference of point A (2.0 cm from the axis of the rod) and point B (4.0 cm from the axis) .

Quiz 20: Electric Potential and Capacitance

For the potential V=3x2z−2yz3V = 3 x ^ { 2 } z - 2 y z ^ { 3 }V=3x2z−2yz3 ,what is the corresponding electric field at the point (2,2,2) ?

A linear charge of nonuniform density λ\lambdaλ = bx,where b = 2.1 nC/m2,is distributed along the x axis from x = 2.0 m to x = 3.0 m.Determine the electric potential (relative to zero at infinity) of the point y = 4.0 m on the y axis.

An electric dipole having dipole moment of magnitude p is placed in a uniform electric field having magnitude E.What is the magnitude of the greatest change in potential energy that can happen for this dipole in this field?

A charge of 20 nC is distributed uniformly along the y axis from y = 0 to y = 4 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +3 m on the x axis?

A charge of uniform density (0.80 nC/m) is distributed along the x axis from the origin to the point x = 10 cm.What is the electric potential (relative to zero at infinity) at a point,x = 18 cm,on the x axis?

A charge of 18 nC is uniformly distributed along the y axis from y = 3 m to y = 5 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +2 m on the x axis?

A charge of 4.0 nC is distributed uniformly along the x axis from x = +4 m to x = +6 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the origin?

Charge of uniform linear density 6.0 nC/m is distributed along the x axis from x = 0 to x = +3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point y = +4 m on the y axis?

A charge per unit length given by λ\lambdaλ (x) = bx,where b = 12 nC/m2,is distributed along the x axis from x = +9.0 cm to x = +16 cm.If the electric potential at infinity is taken to be zero,what is the electric potential at the point P on the y axis at y = 12 cm?

A rod (length = 2.0 m) is uniformly charged and has a total charge of 5.0 nC.What is the electric potential (relative to zero at infinity) at a point which lies along the axis of the rod and is 3.0 m from the center of the rod?

Charge of uniform linear density 3.0 nC/m is distributed along the x axis from x = 0 to x = 3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +4 m on the x axis?

A charge of 20 nC is distributed uniformly along the x axis from x = -2.0 m to x = +2.0 m.What is the electric potential (relative to zero at infinity) at the point x = 5.0 m on the x axis?

A charge of 10 nC is distributed uniformly along the x axis from x = -2 m to x = +3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +5 m on the x axis?

Charge of uniform density 12 nC/m is distributed along the x axis from x = 2.0 m to x = 5.0 m.What is the electric potential (relative to zero at infinity) at the origin (x = 0) ?

The electric field in the region defined by the y-z plane and the negative x axis is given by E = -ax,where a is a constant.(There is no field for positive values of x. ) As -x increases in magnitude,relative to V = 0 at the origin,the electric potential in the region defined above is

Which of the following represents the equipotential lines of a dipole?

Charge of uniform density (3.5 nC/m) is distributed along the circular arc shown.Determine the electric potential (relative to zero at infinity) at point P.

A nonuniform linear charge distribution given by λ\lambdaλ (x) = bx,where b is a constant,is distributed along the x axis from x = 0 to x = +L.If b = 40 nC/m2 and L = 0.20 m,what is the electric potential (relative to a potential of zero at infinity) at the point y = 2L on the y axis?

The electric field in the region defined by the y-z plane and the positive x axis is given by E = ax,where a is a constant.(There is no field for negative values of x. ) As x increases in magnitude,relative to V = 0 at the origin,the electric potential in the region defined above is

Charge of uniform density 50 nC/m3 is distributed throughout the inside of a long nonconducting cylindrical rod (radius = 5.0 cm) .Determine the magnitude of the potential difference of point A (2.0 cm from the axis of the rod) and point B (4.0 cm from the axis) .

For the potential $V = 3 x ^ { 2 } z - 2 y z ^ { 3 }$ ,what is the corresponding electric field at the point (2,2,2) ?

A linear charge of nonuniform density $\lambda$ = bx,where b = 2.1 nC/m²,is distributed along the x axis from x = 2.0 m to x = 3.0 m.Determine the electric potential (relative to zero at infinity) of the point y = 4.0 m on the y axis.

A charge per unit length given by $\lambda$ (x) = bx,where b = 12 nC/m²,is distributed along the x axis from x = +9.0 cm to x = +16 cm.If the electric potential at infinity is taken to be zero,what is the electric potential at the point P on the y axis at y = 12 cm?

A nonuniform linear charge distribution given by $\lambda$ (x) = bx,where b is a constant,is distributed along the x axis from x = 0 to x = +L.If b = 40 nC/m² and L = 0.20 m,what is the electric potential (relative to a potential of zero at infinity) at the point y = 2L on the y axis?

Charge of uniform density 50 nC/m³ is distributed throughout the inside of a long nonconducting cylindrical rod (radius = 5.0 cm) .Determine the magnitude of the potential difference of point A (2.0 cm from the axis of the rod) and point B (4.0 cm from the axis) .