Assignment 1, solutions

Problem 1:

imageConsider the circuit shown.  At t = 0 the switch is closed.
(a)  What are the values of the currents through the resistors R1, R2, and R3 just after the switch has been closed and a long time after the switch has been closed?
(b)  Find the currents through the resistors R1, R2, and R3 as a function of time for t > 0.

Solution:

Problem 2:

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.

image
(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 v0 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, mv02/2.  Where does this energy go?  Prove that energy is conserved in this process by showing that the energy gained elsewhere is exactly mv02/2.

Solution:

Problem 3:

imageA 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 = B0 (1 + kx), where B0 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.

Solution:

Problem 4:

imageIn 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 V0 and a 120 cycle component of amplitude V2, find the corresponding voltages at the terminals of the capacitor for R = 103 Ω and C = 10 μF.

Solution:

Problem 5:

imageRefer 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?

Solution: