Interactive Electric Plane

An infinite plane generates a uniform electric field of magnitude \(E = 80 \, \text{N/C} \,\) 3 m away from itself.

a) What is the magnitude of the electric field if the distance from the plane is doubled?

b) What is the magnitude of the electric field if the distance from the plane is halved?

Question 1 of 14

1. Before you start, please read the following instructions: First, read the prompt of the problem carefully. Second, we recommend that you take your own notes as you move through the questions. Enjoy!

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Question 1 of 14

Question 2 of 14

2. Select all the variables of the problem that are known

The magnitude of the electric field 3 and 6 meters away from the plane, and the charge density of the plane

The magnitude of the electric field 3 meters away from the plane and the charge density of the plane

The magnitude of the electric field 3 meters away from the plane and the area of the plane

The electric field 3 meters away from the plane and the area of the plane

Question 2 of 14

Question 3 of 14

3. Which of the following strategies you think will be more useful for solving the problem:

Find an equation for the total Coulomb force that an infinite plane produces at an arbitrary distance.

Find an equation for the electric field of an infinite plane at an arbitrary distance.

Find an equation for the electric field of an infinite plane 3 meters away from the plane.

Find an equation for the work required to move a point particle with charge \(q\) from infinity to a distance \( d\) to the plane

Question 3 of 14

Question 4 of 14

4. In order to find a general expression for the electric field produced by the plane at an arbitrary distance, we can use Gauss' law.

True

False

Question 4 of 14

Question 5 of 14

5. Which of the following is Gauss’ law:

\( \oint_S \vec{E}\cdot d\vec{A}=\frac{Q_{\text{enc}}}{\epsilon_0} \)

\( \vec{E} \cdot \oint_S dA=\frac{Q_{\text{enc}}}{\epsilon_0} \)

\( \oint_S E dA=\frac{Q_{\text{enc}}}{\epsilon_0} \)

\( \oint_S \vec{E} d\vec{A}=\frac{Q_{\text{enc}}}{\epsilon_0} \)

Question 5 of 14

Question 6 of 14

6. Which of the following statements better describe Gauss’ law:

The magnitude of the electric field produced by a physical system is given by the total charge of that system

The electric field produced by a physical system is given by the total charge of that system.

The total electric flux through a physical system is given by the charge inside of that system.

The total electric flux through a Gaussian surface is given by the charge enclosed by the surface.

Question 6 of 14

Question 7 of 14

7. In order to apply Gauss’ law, we must take into consideration the shape of the system because…

this shape will give information about the total area of the Gaussian surface

this shape will give information about the symmetries of the electric field produced by the system

this shape will give information about whether the electric field is positive or negative

this shape will give information about the charge density

Question 7 of 14

Question 8 of 14

8. Which of the following better represents some of the lines for the electric field produced by this infinite plane:

Question 8 of 14

Question 9 of 14

9. Given the symmetries of the electric field for the plane, we should use the following Gaussian surface:

Question 9 of 14

Question 10 of 14

10. Write the integral for Gauss' law explicitly, taking into account all the faces of the Gaussian surface. Which of the following expressions is equivalent to your result:

\( \oint_S\vec{E}\cdot d\vec{A}=\int_{S_1}\vec{E}_1\cdot d\vec{A}_1+\int_{S_2}\vec{E}_2\cdot d\vec{A}_2+\int_{S_3}\vec{E}_3\cdot d\vec{A}_3 \)

\( \oint_S\vec{E}\cdot d\vec{A}=\int_{S_1}\vec{E}_1\cdot d\vec{A}_1 \)

\( \oint_S\vec{E}\cdot d\vec{A}=\int_{S_1}\vec{E}_1\cdot d\vec{A}_1+\int_{S_2}\vec{E}_2\cdot d\vec{A}_2 \)

Question 10 of 14

Question 11 of 14

11. Perform the integration for Gauss' law and solve for the electric field in terms of the enclosed charge \( Q_{enc} \) and the area \(A\) of the top and bottom faces. Which of the following expressions is equivalent to your result?

\( E=\frac{Q_{enc}}{A\epsilon_0} \)

\( E=\frac{Q_{enc}}{2A} \)

\( E=\frac{Q_{enc}}{2\epsilon_0} \)

\( E=\frac{Q_{enc}}{2A\epsilon_0} \)

Question 11 of 14

Question 12 of 14

12. Suppose that we want our final answer to not depend on the area \(A\) of the top and bottom surfaces but only on the charge density. In that case, we should

Write \(A\) on the left side as \( \pi r^2\) , where \(r\) is the radius of the top and bottom faces of the cylinder.

Write the charge \(Q_{enc}\) in terms of \(\sigma\), the charge density per unit of area.

Write the charge \(Q_{enc}\) in terms of \(\rho\), the charge density per unit of volume.

Write \(A\) on the left side of the expression for Gauss' law in terms of the charge density, and the charge \(Q_{enc}\) on the right side of the expression in terms of the charge density as well so that these densities cancel out.

Question 12 of 14

Question 13 of 14

13. According to the expression for \( E \) found before, we can say that

The magnitude of the electric field for an infinite plane does not depend on the distance to the plane.

The electric flux for an infinite plane does not depend on the distance to the plane.

The magnitude of the electric field for an infinite plane does not depend on the area of the plane.

The magnitude of the electric field for an infinite plane does not depend on the charge density of the plane.

Question 13 of 14

Question 14 of 14

14. The final answer for (a) and (b) is

We still need more information

80 N/C and 40 N/C respectively.

80 N/C and 80 N/C, the same.

80 N/C and 160 respectively

Question 14 of 14