32. Electromagnetic Waves

The magnetic field within a long, straight solenoid with a circular cross section and radius R is increasing at a rate of dB/dt. (a) What is the rate of change of flux through a circle with radius r₁ inside the solenoid, normal to the axis of the solenoid, and with center on the solenoid axis? (b) Find the magnitude of the induced electric field inside the solenoid, at a distance r₁ from its axis. Show the direction of this field in a diagram. (c) What is the magnitude of the induced electric field outside the solenoid, at a distance r₂ from the axis?

A 2.0 cm×2.0 cm square loop of wire with resistance 0.010 Ω has one edge parallel to a long straight wire. The near edge of the loop is 1.0 cm from the wire. The current in the wire is increasing at the rate of 100 A/s. What is the current in the loop?

You measure the electric field of an electromagnetic wave at a particular moment and find it points in the +z direction. The magnetic field points in the +y direction. In which direction is this wave traveling?

You measure the magnetic field strength of a traveling electromagnetic wave to be $8.0\times10^{-7}T$ , oriented along the +x direction. If this EM wave moves in the +y direction, what is the magnitude and direction of the wave's electric field at that same exact spot?

CALC An electric generator has an 18-cm-diameter, 120-turn coil that rotates at 60 Hz in a uniform magnetic field that is perpendicular to the rotation axis. What magnetic field strength is needed to generate a peak voltage of 170 V?

CALC The rectangular loop in FIGURE CP30.81 has 0.020 Ω resistance. What is the induced current in the loop at this instant?

One way to measure the strength of a magnetic field is with a flip coil. Suppose a 200-turn, 4.0-cm-diameter coil with a resistance of 2.0 Ω is connected to a ballistic galvanometer, a device that measures the total charge passing through. The coil is held perpendicular to the field, then quickly flipped 180° so that the opposite side is facing the magnetic field. Afterward, the galvanometer reads 7.5 μC. What is the field strength? Hint: Use I = dq/dt to relate the net change of flux to the amount of charge that flows through the galvanometer.

It has been proposed that small spacecraft could reach other planets in a fairly short time—days instead of many months—if they unfurl a reflective sail and are accelerated by a powerful laser beam generated by an earth-orbiting laser. What speed would a spacecraft need to reach Mars in 8.0 days when Mars is closest to earth?

(II) Our nearest star (other than the Sun) is 4.2 light-years away. That is, it takes 4.2 years for the light it emits to reach Earth. How far away is it in meters?

(II) A light-year is a measure of distance (not time). How many meters does light travel in a year?

Assume that the nearest stars to us have an intrinsic luminosity about the same as the Sun’s. Their apparent brightness, however, is about 10¹¹ times fainter than the Sun. From this, estimate the distance to the nearest stars.

(II) An electromagnetic wave has an electric field given by $\vec{E}=\hat{i}\left(2775\text{ V/m}\right)\sin \left[\right(0.053{\text{ m}}^{-1})z-(1.6\times {10}^7\text{ rad/s}\left)t\right]$.

a. What are the wavelength and frequency of the wave?

b. Write down an expression for the magnetic field.