Two long, thin cylindrical conducting shells of radii r_{1}
and r_{2}, respectively, are oriented coaxially (one cylinder is
centered inside the other). The inside cylinder with radius r_{1}
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?

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** = -E_{0} **k**.

The displacement vector from electric dipole **p**_{1} to dipole **p**_{2} is
**r**.

(a) Calculate the electric potential energy W;

(b) Calculate the force **F**_{21} that **p**_{1} exerts on
**p**_{2}.

(c) Calculate the torque **τ**_{12} that **p**_{1} exerts on
**p**_{2}.

(d) Let **p**_{1} = (10^{-9} Cm)**k** be located at
the origin and **p**_{2} = (10^{-9} Cm)**i **at **r** =
(3 m)**i** + (4 m)**k**.

Provide numeric answers for **F**_{21} and **τ**_{12}.

Suppose the input voltages V_{1}, V_{2} and V_{3} 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 V_{out} for each of these possibilities.

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.

(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/(ax^{2} + bx + c)^{½} = (ax^{2}
+ bx + c)^{½}/a - (b/(2a))∫ dx/(ax^{2} + bx + c)^{½
}∫ dx/(ax^{2} + bx + c)^{½} = a^{-½} ln(2a^{ ½}
(ax^{2} + bx + c)^{½} + 2ax + b)