# Physics 3211: Electromagnetic Theory (Tutorial)

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1 Question 1 a) The capacitor shown in Figure 1 consists of two parallel dielectric layers and a voltage source, V. Derive an equation for capacitance. b) Find the capacitance for the configuration of Figure 1 such that the plates are on the left and right faces of the structure.

2 Question 2 A coaxial cable consists of an inner cable made from a perfect conductor, having radius a, and surrounded by tubular perfect conductor of inner radius b>a. The outer radius of the tube is B. If the space between the inner and outer conductors is filled with air (ε= ε 0 ), as illustrated on the left of Figure 1 and if the outer conductor is electrically neutral and if a net static charge +Q C/m is supplied to the inner cable: a) Find the E-field: i. Inside the inner cable, ii. Between the two conductors, iii. Inside the outer tubular conductor and, iv. Outside the arrangement. b) Indicate where there are charges and find the corresponding charge densities. c) Find the capacitance per unit length of this device.

3 Question 3 An infinitely long thin wire arriving from along ŷ is bent in a semicircle of radius a centered at the origin, and returns to along the line y = a. a) If the wires carry a uniform linear charge density λ 1 C/m, find the electric field at the origin. b) If an infinitely long wire with linear charge density λ 2 is available, how must it be placed so that the net electric field of the bent wire and the infinite wire is zero at the origin?

4 Question 4 Indicate if the following statements are true or false. Justify briefly your answers. a) The electrostatic potential V is a vector because current flows from the to the + terminal of a battery. b) Gauss law holds only if the charge distribution has a specific symmetry. c) In electrostatics, Gauss law holds only for continuous charge distributions but not for point charges. Question 5 A Semi-infinite wire lies on the negative z axis, from z = 0 to z =, with constant linear charge density λ. a) Determine E at a point (0, 0, z) on the positive z axis. b) Determine E at a point (x, 0, 0) on the positive x axis.

5 Question 6 The figure below shows a conducting sphere of radius r 1 surrounded by a thick conducting shell of inner radius r 2 and outer radius r 3. The region between the sphere and the shell is filled with an electrically neutral dielectric characterized by ε. A net charge of Q is located on the inside sphere. a) Find an expression for E valid for any point R p located i) inside the sphere ii) between the conductors iii) inside the thick shell iv) outside the arrangement. b) What is the surface charge density on the inner and outer surfaces of the thick shell? c) What is the potential difference between the shell and the sphere? d) What is the capacitance of this device?

6 Question 7 A wire, initially carrying a current I 1 in a wire placed along the line y=x, is bent into a quarter circle of radius r. The current returns on a straight wire placed along the line y = -x. A second wire is placed parallel to the x-axis, a distance r above the origin. a) Find the field B (magnitude and direction) at the origin due to the bent wire alone. b) What must be the magnitude and direction of the current in the straight wire to have B=0 at the origin?

7 Question 8 A long string of wire is coiled, snake wise, so that it forms a flat disk in the xy plane, containing a very large uniform number N of turns per unit distance along its radius. The coil is centered at the origin, such that the inner radius is a and the outer radius is b, and the wire is made to carry a current I going in the counterclockwise direction. a) Find the direction of the B field in the region r<a where r 2 =x 2 +y 2. b) Find the magnitude of the B field at the origin.

8 Question 9 A Horizontal strip made of non-magnetic material, having width d and extending from y = d/2 to y = -d/2 in the y direction, infinitely long in the x direction and having negligible thickness in the z direction, lying = xj ˆ in air in the x y plane. It has a current density s 0. a) Find B at the point P = (0, 0, z 0 ), where z 0 > 0. J Question 10 Two infinite parallel wires are separated by a distance d carrying equal currents I in opposite directions, with I increasing at the rate di/dt. A square loop of wire of length d on a side lies in the plane of the wires at a distance d from one of the parallel wires. a) Find the emf induced in the square loop. b) Is the induced current clockwise or counterclockwise? Justify your answer.

9 Question 11 The loop shown moves away from a wire carrying a current I 1 =10A at a constant velocity u = yˆ5 ( m/ s) If R=10 Ω and the direction of I 2 is defined in the figure, find I 2 as a function of y 0, the distance between the wire and the loop. Ignore the internal resistance of the loop.

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