Consider the circuit shown. At t = 0 the switch is closed.

(a) What are the values of the currents through the resistors R_{1}, R_{2},
and R_{3} just after the switch has been closed and a long time after
the switch has been closed?

(b) Find the currents through the resistors R_{1}, R_{2}, and R_{3}
as a function of time for t > 0.

A metal bar of mass m slides without friction on two parallel conducting
rails a distance l apart (see figure). A resistor R is connected across
the rails and a uniform magnetic field **B**, pointing into the page, fills
the entire region.

(a) If the bar moves to the right at speed v, what is the current in the
resistor? Indicate also in what direction the current flows.

(b) What is the magnetic force on the bar? Indicate the direction of the
force and provide the result in terms of the data given.

(c) If the bar starts out with speed v_{0} at time t = 0, and is left to
slide, what is its speed at a later time t?

(d) The initial kinetic energy of the bar was, of course, mv_{0}^{2}/2.
Where does this energy go? Prove that energy is conserved in this process
by showing that the energy gained elsewhere is exactly mv_{0}^{2}/2.

A^{ }square loop made of wire with negligible resistance is placed^{
}on a horizontal frictionless table as shown (top view). The^{ }
mass of the loop is m and the length of^{ }each side is b. A
non-uniform vertical magnetic field exists^{ }in the region; its
magnitude is given by the formula^{ }B = B_{0} (1 + kx), where B_{0}
and k are known constants.^{ }

The loop is^{ }given a quick push with an initial velocity v along^{
}the x-axis as shown. The loop stops after a time^{ }interval t.
Find the self-inductance L of the loop.^{ }

In order to suppress the 120 Hz hum from the power supply rectifier or
amplifier, a "smoothing filter" is used. In its simplest form it
consists of a resistance in series with a capacitance, as shown in the figure.
If the applied voltage has a DC component V_{0} and a 120 cycle
component of amplitude V_{2}, find the corresponding voltages at the
terminals of the capacitor for R = 10^{3 }Ω and C = 10 μF.

Refer to the figure. One end of a conducting rod rotates with angular
velocity ω in a circle of radius a making contact with a horizontal, conducting
ring of the same radius. The other end of the rod is fixed. Stationary
conducting wires connect the fixed end of the rod (A) and a fixed point on the
ring (C) to either end of a resistance R. A uniform vertical magnetic field
**B** passes through the ring.

(a) Find the current I flowing through the resistor and the rate at which
heat is generated in the resistor.

(b) What is the sign of the current, if positive I corresponds to flow in the
direction of the arrow in the figure?

(c) What torque must be applied to the rod to maintain its rotation at the
constant angular rate ω?

What is the rate at which mechanical work must be done?