We have a large copper plate with uniform surface charge density, $\sigma$.
Imagine the Gaussian surface drawn below. Calculate the E-field a small distance $s$ above the conductor surface.
<img src="./images/copper_plate.png" align="left" style="width: 300px";/>
1. $|E| = \frac{\sigma}{\varepsilon_0}$
2. $|E| = \frac{\sigma}{2\varepsilon_0}$
3. $|E| = \frac{\sigma}{4\varepsilon_0}$
4. $|E| = \frac{1}{4\pi\varepsilon_0}\frac{\sigma}{s^2}$
5. $|E| = 0$
Note:
* CORRECT ANSWER: A
* Might have to do derivation and go through details of infinitely thin line charge. Must be +sigma on other side, btw.
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A positive charge ($q$) is outside a metal conductor with a hole cut out of it at a distance $a$ from the center of the hole. What is the *net* electric field at center of the hole?
1. $\dfrac{1}{4 \pi \varepsilon_0}\dfrac{q}{a^2}$
2. $\dfrac{-1}{4 \pi \varepsilon_0}\dfrac{q}{a^2}$
3. $\dfrac{1}{4 \pi \varepsilon_0}\dfrac{2q}{a^2}$
4. $\dfrac{-1}{4 \pi \varepsilon_0}\dfrac{2q}{a^2}$
5. Zero
Note: Correct Answer E
With $\nabla \times \mathbf{E} = 0$, we know that,
$$\oint \mathbf{E} \cdot d\mathbf{l} = 0$$
If we choose a loop that includes a metal and interior vacuum (i.e., both in and **inside the hole**), we know that the contribution to this integral in the metal vanishes. What can we say about the contribution in the hole?
1. It vanishes also
2. $\mathbf{E}$ must be zero in there
3. $\mathbf{E}$ must be perpendicular to d$\mathbf{l}$ everywhere
4. $\mathbf{E}$ is perpendicular to the metal surface
5. More than one of these
Note:
* Correct answer: E (A and B)
With $\nabla \times \mathbf{E} = 0$, we know that,
$$\oint \mathbf{E} \cdot d\mathbf{l} = 0$$
If we choose a loop that includes a metal and vacuum (i.e., both in and **just outside of the metal**), we know that the contribution to this integral in the metal vanishes. What can we say about the contribution just outside the metal?
1. It vanishes also
2. $\mathbf{E}$ must be zero out there
3. $\mathbf{E}$ must be perpendicular to d$\mathbf{l}$ everywhere
4. $\mathbf{E}$ is perpendicular to the metal surface
5. More than one of these
Note:
* Correct answer: E (both A and D)
A neutral copper sphere has a spherical hollow in the center. A charge $+q$ is placed in the center of the hollow. What is the total charge on the outside surface of the copper sphere? (Assume Electrostatic equilibrium.)
<img src="./images/coppersphere_hole_and_charge.png" align="left" style="width: 350px";/>
1. Zero
2. $-q$
3. $+q$
4. $0 < q_{outer} < +q$
5. $-q < q_{outer} < 0$
Note:
* Correct answer: C
<img src="./images/cylinder_charge_outside.png" align="right" style="width: 250px";/>
A long coax has total charge $+Q$ on the OUTER conductor. The INNER conductor is neutral.
What is the sign of the potential difference, $\Delta V = V(c)-V(0)$, between the center of the inner conductor ($s = 0$)
and the outside of the outer conductor?
1. Positive
2. Negative
3. Zero
Note:
* CORRECT ANSWER: C; there's no field inside at all
<img src="./images/conducting_cap_plates_simple.png" align="right" style="width: 300px";/>
Given a pair of very large, flat, conducting capacitor plates with total charges $+Q$ and $-Q$. Ignoring edges, what is the equilibrium distribution of the charge?
1. Throughout each plate
2. Uniformly on both side of each plate
3. Uniformly on top of $+Q$ plate and bottom of $–Q$ plate
4. Uniformly on bottom of $+Q$ plate and top of $–Q$ plate
5. Something else
Note:
* CORRECT ANSWER: D
<img src="./images/conducting_cap_plates.png" align="right" style="width: 400px";/>
Given a pair of very large, flat, conducting capacitor plates with surface charge densities $+/-\sigma$, what is the E field in the region between the plates?
1. $\sigma/2\varepsilon_0$
2. $\sigma/\varepsilon_0$
3. $2\sigma/\varepsilon_0$
4. $4\sigma/\varepsilon_0$
5. Something else
Note:
* CORRECT ANSWER: B