Assignment 1

Problem 1:

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?

Problem 2:

(a)  Find the magnetic field at any point in the x-y plane for y > 0 due to a wire of length l carrying a current I from x = 0 to x = l along the x-axis.
(b)  Use the result of (a) to find the self inductance of a square current loop of side l.  You may leave your result as a definite integral, but the limits must be specified and the integrand must be in terms of the dimensions of the loop, constants, and the variables being integrated.
(c)  A conducting square current loop of side l with sides parallel to the x- and y-axis has resistance R and self inductance L.  It moves at a speed v in the +x-direction.  The loop passes through a magnetic field given by B = B0 k, -l/2< x < l/2, B = 0 elsewhere.  Find the current in the wire as a function of time assuming that I = 0 a long time before the loop reaches the magnetic field.

Problem 3:

Consider the impedance bridge shown in the figure below.  Its purpose is to permit measurements of an unknown impedance Zu in terms of the fixed resistance RA and RB, variable resistance RS and variable capacitance CS.  If Zu is a pure resistance, then CS may be removed from the circuit (shorted out) and the impedance bridge becomes a simple Wheatstone bridge.
(a)  For a purely resistive impedance Zu find the value of RS for which a balance is obtained (no current on the null detector).
(b)  For a complex impedance Zu whose resistive component is Ru and whose reactive component is Xu, find the values of RS and CS for which balance is obtained.  Assume Zu has zero inductance. 
(c)  Show that for a purely inductive component Zu balance is not possible.

Problem 4:

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 = 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.

Problem 5:

A DC motor has coils with a resistance of 30 Ω and operates from a voltage of 240 V.  When the motor is operating at its maximum speed, the peak back emf is 145 V.  
Find the peak current in the coils
(a)  when the motor is first turned on and
(b)  when the motor has reached maximum speed.
(c)  If the current in the motor is 6.0 A at some instant, what is the back emf at that time?