Physics 482

The current in an infinite solenoid of radius $R$ with uniform magnetic field $\mathbf{B}$ inside is increasing so that the magnitude $B$ in increasing with time as $B=B_0+kt$. If I calculate $V$ along path 1 and path 2 between points A and B, do I get the same answer? <img src="./images/V_outside_solenoid.png" align="left" style="width: 500px";/> 1. Yes 2. No 3. Need more information Note: * Correct Answer: B * My explanation involves going to the case with NO solenoid, where we know that the integral from A to B is path independent, and thus the loop integral all the way around is zero. So, A to B and B to A (the other path) CANCEL each other, making A to B and A to B (other path) EQUAL each other. That’s when the line integral IS zero. But now the line (loop) integral is NOT zero, and so those two integrals cannot still cancel.

## Announcements * Homework 4 is posted * Due in class on Monday! * Project problem - starting to write your paper (give yourself time to write). * Quiz 2 - This Friday (Motional EMF) * Discuss the differences between: * $\mathcal{E} = \oint \mathbf{f} \cdot d\mathbf{l}$ and $\mathcal{E} = -\frac{d\Phi_B}{dt}$ * Solve a motional EMF problem and discuss the direction of the current

The current in an infinite solenoid with uniform magnetic field $\mathbf{B}$ inside is increasing so that the magnitude $B$ in increasing with time as $B=B_0+kt$. A small circular loop of radius $r$ is placed coaxially inside the solenoid as shown. Without calculating anything, determine the direction of the induced magnetic field created by the induced current in the loop, in the plane region inside the loop? <img src="./images/loop_at_center_of_solenoid.png" align="left" style="width: 300px";/> 1. Into the screen 2. Out of the screen 3. CW 4. CCW 5. Not enough information Note: * Correct Answer: A

The current in an infinite solenoid with uniform magnetic field $\mathbf{B}$ inside is increasing so that the magnitude B is increasing with time as $B=B_0+kt$. A circular loop of radius $r$ is placed coaxially outside the solenoid as shown. In what direction is the induced $\mathbf{E}$ field around the loop? <img src="./images/solenoid_w_loop_outside.png" align="left" style="width: 300px";/> 1. CW 2. CCW 3. The induced E is zero 4. Not enough information

A long solenoid of cross sectional area, $A$, creates a magnetic field, $B_0(t)$ that is spatially uniform inside and zero outside the solenoid. SO: <img src="./images/solenoid_with_B_shown.png" align="center" style="width: 600px";/> 1. $E=\dfrac{\mu_0 I}{2 \pi r}$ 2. $E=-A\dfrac{\partial B}{\partial t}\dfrac{1}{\pi r^2}$ 3. $E=-A2\pi r\dfrac{\partial B}{\partial t}$ 4. $E=-A \dfrac{\partial B}{\partial t}\dfrac{1}{2 \pi r}$ 5. Something else Note: * Correct Answer: D

If the arrows represent an E field, is the rate of change in magnetic flux (perpendicular to the page) through the dashed region zero or nonzero? <img src="./images/curly_E_1.png" align="right" style="width: 500px";/> 1. $\frac{d\Phi}{dt} = 0$ 2. $\frac{d\Phi}{dt} \neq 0$ 3. ??? Note: * Correct Answer: A * Curl E is zero everywhere except at the origin! So, if our loop enclosed the origin, we'd be in trouble!

If the arrows represent an E field (note that |E| is the same everywhere), is the rate of change in magnetic flux (perpendicular to the page) in the dashed region zero or nonzero? <img src="./images/curly_E_2.png" align="right" style="width: 500px";/> 1. $\frac{d\Phi}{dt} = 0$ 2. $\frac{d\Phi}{dt} \neq 0$ 3. Need more information Note: * Correct Answer: B

The current $I_1$ in loop 1 is increasing. What is the direction of the induced current in loop 2, which is co-axial with loop 1? <img src="./images/mutual_coil_1.png" align="center" style="width: 300px";/> 1. The same direction as $I_1$ 2. The opposite direction as $I_1$ 3. There is no induced current 4. Need more information Note: * Correct Answer: B

The current $I_1$ in loop 1 is increasing. What is the direction of the induced current in loop 2, which lies in the same plane as loop 1? <img src="./images/mutual_coil_2.png" align="center" style="width: 500px";/> 1. The same direction as $I_1$ 2. The opposite direction as $I_1$ 3. There is no induced current 4. Need more information Note: * Correct Answer: A

The current $I_1$ in loop 1 is decreasing. What is the direction of the induced current in loop 2, which lies in a plane perpendicular to loop 1 and contains the center of loop 1? <img src="./images/mutual_coil_3.png" align="center" style="width: 400px";/> 1. The same direction as $I_1$ 2. The opposite direction as $I_1$ 3. There is no induced current 4. Need more information Note: * Correct Answer: C

Two flat loops of equal area sit in a uniform field $\mathbf{B}$ which is increasing in magnitude. In which loop is the induced current the largest? (The two wires are insulated from each other at the crossover point in loop 2.) <img src="./images/mutual_coil_4.png" align="center" style="width: 600px";/> 1. Loop 1 2. Loop 2 3. They are both the same 4. Not enough info Note: * Correct Answer: A

<img src="./images/solenoid_w_loop_mutual.png" align="right" style="width: 300px";/> A loop of wire 1 is around a very long solenoid 2. <img src="./images/phi_1_eqn.png" align="left" style="width: 200px";/> = the flux through loop 1 due to the current in the solenoid <img src="./images/phi_2_eqn.png" align="left" style="width: 200px";/> = the flux through the solenoid due to the current in loop 1 Which is easier to compute? 1. $M_{12}$ 2. $M_{21}$ 3. equally difficult to compute Note: * Correct Answer: A