Q: At $$z = 0$$ and $$t = 0$$, $$|B_x(0,0)| = 100 T. $$What is $$|E_y(0,0)| at $$the same place and time? Use $$c = 2.99792458 \times 10^8$ m/s.

$$|E_y(0,0)| = 2.99792458 \times 10$$^{10}$$ V/m

Question 1

The magnetic field of an electromagnetic wave is given by $$\mathbf{B}(z, t) = \begin{pmatrix} B_x(z, t) \ 0 \ 0 \end{pmatrix}. If at $$some time and position the magnetic field points along the positive $$z-$$axis, in which direction does the electric field point at the same time and position?

Accepted Answer

Along the positive $$y-$$axis

Question 2

For the choice of electric field direction that you made above, in which direction does the electromagnetic wave propagate?

Accepted Answer

Along the positive $$z-$$axis

Question 3

Assume that $$B_x(z, t)$$ has its maximum positive value at $$z = 0$$ and $$t = 0. $$Calculate the location of its maximum positive value (that is, the $$z-$$value at which it occurs) at time $$t = 10$$^{-8}$$ s. Use c = 2.99792458$$ 	imes $$10^8 m/s.$$

Accepted Answer

$$z = 2.99792458 m$$

Question 4

At $$z = 0$$ and $$t = 0$$, $$|B_x(0,0)| = 100 T. $$What is $$|E_y(0,0)| at $$the same place and time? Use $$c = 2.99792458 	imes 10^8$ m/s.

Accepted Answer

$$|E_y(0,0)| = 2.99792458 	imes 10$$^{10}$$ V/m

Question 5

You have a circular loop of wire. As you rotate it, $$\vec{A} is $$perpendicular to the plane that contains the maximum flux of the magnetic field through the loop. What must be true for the maximum amount of current to be generated in the loop because of the presence of the electromagnetic wave?

Accepted Answer

The loop must be perpendicular to the direction of the magnetic field, and the magnetic field must be time-dependent

Question 6

A coil is moving with velocity $$v = 50 cm/s $$into a region with magnetic field $$B = 2 T. $$The coil has length $$a = 8 cm. $$What is the current in coil 1 as it enters the magnetic field region?

Accepted Answer

$$I = \frac{aBv}{R} = 0.08 A$$, clockwise

Question 7

What is the current in coil 2, right after its leading edge has entered the magnetic field?

Accepted Answer

The current is zero, because the contributions of both field regions to the flux cancel when they are added together to calculate the magnetic flux.

Question 8

What is the current in coil 2, when it overlaps both magnetic field regions?

Accepted Answer

$$I = -2\frac{aBv}{R} = -0.16 A$$, counterclockwise

Question 9

A solenoid is connected to a voltage source providing a potential of $$V_0 = 10$ V. The switch has been closed for a long time, and then is suddenly opened at time t = 0$. What is the current I_1$ passing through the solenoid immediately after the switch is opened?

Accepted Answer

$$I_1$$ = \frac{$$V_0$$}{2R}$$

Question 10

A solenoid is connected to a voltage source providing a potential of $$V_0 = 10$ V. The switch has been closed for a long time, and then is suddenly opened at time t = 0$. What is the current I_2$ passing through the solenoid a long time after the switch has been opened?

Accepted Answer

$$I_2 = 0$

Question 11

The time-dependence of the current in the solenoid after the switch has been opened can be written as $$I(t) = A \exp(-t/	au) + B. $$What is the algebraic expression for $$	au as a $$function of $$R$$, $$L$$, and $$N$$?

Accepted Answer

$$	au = \frac{L}{2R}$$

Question 12

What is the electric field in the exact center of the solenoid immediately after the switch is opened?

Accepted Answer

Even though the magnetic field changes, the electric field is zero here, by symmetry.

Question 13

Consider the circuit depicted in the figure. A voltage supply with $$V = 12 V is $$connected with a capacitor $$C$$ and two equal resistors with $$R = 10 \Omega. $$What is the current $$I_1$$ passing through the outer resistor?

Accepted Answer

$$I_1$$ = \frac{V}{2R} = 0.6$$ A

Question 14

What is the charge on the capacitor in the circuit above? (Algebraic expression and numerical value)

Accepted Answer

$$Q = CR_1$ \frac{V}{3R} = 8 \mu C$$

Question 15

In the circuit with a capacitor and two resistors, what is the current $$I_2$$ passing through the resistor in parallel with the capacitor?

Accepted Answer

$$I_2$$ = \frac{V}{3R} = 0.4$$ A

Question 16

A capacitor is charged to a voltage $$V_C. $$What is the energy stored in the capacitor?

Accepted Answer

$$U = \frac{1}{2}CV_C^2$

Question 17

A solenoid of radius $$r = 1 cm $$and length $$l = 50 cm is $$connected to a voltage source $$V_0 = 10$ V. What is the magnetic field at the center of the solenoid immediately after the switch is opened? Use N = 5000$ windings.

Accepted Answer

$$B = \frac{\mu_0$ N I}{l}$$

Electromagnetic Waves, Magnetic Fields, and Circuits: Study Notes

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