Assignment 2

Problem 1:

Two long, thin cylindrical conducting shells of radii r1 and r2, respectively, are oriented coaxially (one cylinder is centered inside the other).  The inside cylinder with radius r1 carries a linear charge density λ while the outside cylinder carries an equal and opposite linear charge density -λ.
(a)  In what direction does the electric field point between the two cylinders?  Explain.
(b)  Find the electric field
        (i)  inside the smaller cylinder,
        (ii)  between the cylinders,
        (iii)  outside the cylinders.
(c)  Plot the total electric field as a function of r, i.e., as a function of the distance from the cylinders' center.
(d)  Find the potential difference between the two cylinders.
(e)  The construction defines a coaxial cable which is a device useful to transmit information.  Find the capacity per unit length of this cable assuming that there is vacuum between the two cylinders.
(f)  Now assume that keeping the charge of the cylinders constant the space between them is filled with a dielectric with dielectric constant κ.  Does the capacity per unit length change?  Why?

Problem 2:

Find the energy necessary to move a charge Q from point A = (a,0,0) to point B = (a,0,h), where a and h are positive constants, along a helical path parametrized as
r = a cosθ i + a sinθ j + hθ/(2π) k,
under the influence of an electrostatic field E = -E0 k.

Problem 3:

The displacement vector from electric dipole p1 to dipole p2 is r.
(a)  Calculate the electric potential energy W;
(b)  Calculate the force F21 that p1 exerts on p2.
(c)  Calculate the torque τ12 that p1 exerts on p2.
(d)  Let p1 = (10-9 Cm)k be located at the origin and p2 = (10-9 Cm)i at r = (3 m)i + (4 m)k.
Provide numeric answers for  F21 and τ12.

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Problem 4:

Suppose the input voltages V1, V2 and V3 in the circuit shown can assume values of either 0 or 1 (0 means ground).  There are thus 8 possible combinations of input voltages.  List the Vout for each of these possibilities.

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Problem 5:

A conical surface (an empty ice-cream cone) carries a uniform surface charge σ.  The height of the cone is a, as is the radius of the top.
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(a)  Find the electrical potential at point P (vertex of the cone).
(b)  Find the electrical potential at point Q (center of the top of the cone).
(c)  Find the potential difference between points P (the vertex) and Q (the center of the top).

x dx/(ax2 + bx  + c)½ = (ax2 + bx  + c)½/a - (b/(2a))∫ dx/(ax2 + bx  + c)½
∫ dx/(ax2 + bx  + c)½ = a ln(2a ½ (ax2 + bx  + c)½ + 2ax + b)