Q: Given the electric potential of a dipole $$V = ke \frac{p \cos \theta}{r^2$}$$, what is the expression for the $$E_r$$ component of the electric field in spherical coordinates?

$$E_r = ke \frac{2p \cos \theta}{r^3$}$$

Q: For a dipole potential $$V = ke \frac{p \cos \theta}{r^2$}$$, what is the $$E_\theta$$ component of the electric field?

$$E_\theta = ke \frac{p \sin \theta}{r^3$}$$

Question 1

Given the electric potential of a dipole $$V = ke \frac{p \cos 	heta}{r^2$}$$, what is the expression for the $$E_r$$ component of the electric field in spherical coordinates?

Accepted Answer

$$E_r = ke \frac{2p \cos 	heta}{r^3$}$$

Question 2

For a dipole potential $$V = ke \frac{p \cos 	heta}{r^2$}$$, what is the $$E_	heta$$ component of the electric field?

Accepted Answer

$$E_	heta = ke \frac{p \sin 	heta}{r^3$}$$

Question 3

What is the $$E_\phi$$ component of the electric field for the dipole potential $$V = ke \frac{p \cos 	heta}{r^2$}$$?

Accepted Answer

$$E_\phi = 0$$

Question 4

Which symmetry is present in the problem of an infinite uniformly charged plane, and what does it imply about the electric field?

Accepted Answer

Planar symmetry; the electric field is perpendicular to the plane and uniform above and below.

Question 5

State Gauss’s law in equation form and name the quantities involved.

Accepted Answer

$$\int_S \vec{E} \cdot d\vec{A} = \frac{q_{encl}}{\$$epsilon_0$$}$$; $$\vec{E} is $$the electric field, $$d\vec{A} is $$the infinitesimal area vector, $$q_{encl} is $$the enclosed charge, $$\$$epsilon_0$$ is the permittivity of free space.

Question 6

Using Gauss’s law, what is the electric field produced by an infinite plane with charge density $$\sigma$$?

Accepted Answer

$$E = \frac{\sigma}{2\$$epsilon_0$$}$$

Question 7

If a box-shaped Gaussian surface is used to apply Gauss’s law to an infinite plane, what is the total flux through the surface?

Accepted Answer

$$2EA$$

Question 8

What does it mean to connect capacitors in parallel?

Accepted Answer

The same potential difference is applied to each capacitor.

Question 9

What is the formula for the equivalent capacitance of three capacitors $$C_1$$, $$C_2$$, and $$C_3$$ connected in parallel?

Accepted Answer

$$C_{eq} = C_1$ + C_2$ + C_3$

Question 10

When capacitors are connected in parallel, how is the total charge $$Q$$ related to the individual charges $$q_1$$, $$q_2$$, ...?

Accepted Answer

$$Q = q_1$ + q_2$ + ...$$

Question 11

A plane with a disc of radius $$R$$ removed is uniformly charged with density $$\sigma. $$What is the electric field at point $$P on $$the $$z-$$axis above the hole?

Accepted Answer

$$E = \frac{\sigma}{2\$$epsilon_0$$} \frac{z}{\sqrt{z^2$ + R^2$}}$$

Question 12

For a disc of radius $$a$$ and uniform charge density $$\sigma$$, what is the electric field on its axis at distance $$d$$ above the disc?

Accepted Answer

$$E = \frac{\sigma}{2\$$epsilon_0$$} \left(1 - \frac{d}{\sqrt{d^2$ + a^2$}}\right)$$

Question 13

If the electric field above a punctured plane is $$E = \frac{\sigma}{2\$$epsilon_0$$} \frac{z}{\sqrt{z^2$ + R^2$}}$$, what is the direction of the field below the plane?

Accepted Answer

Opposite to the direction above the plane

Question 14

A student claims that the electric field from an infinite plane is $$E = \frac{\sigma}{\$$epsilon_0$$}. $$What is the error in this statement?

Accepted Answer

The correct field is $$E = \frac{\sigma}{2\$$epsilon_0$$}$$; the student forgot to account for the field splitting above and below the plane.

Question 15

Which of the following best describes the use of the superposition principle in finding the electric field of a punctured plane?

Accepted Answer

Subtract the field of the disc from the field of the entire plane.

Question 16

If the potential $$V is $$independent of the $$\phi$$ coordinate, what does this imply about the $$E_\phi$$ component of the electric field?

Accepted Answer

$$E_\phi = 0$$

Question 17

A real-world scenario: In a medical device, three capacitors are connected in parallel to store energy for a defibrillator. If their capacitances are $$C_1 = 2$$ \mu F$$, C_2$ = 3 \mu F$$, and $$C_3 = 5$$ \mu F$$, what is the total capacitance?

Accepted Answer

$$C_{eq} = 10 \mu F$$

Electric Fields, Gauss' Law, and Capacitance: Step-by-Step Physics Guidance

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