Find an expression for the electric field as a function of

Question

Question 1. A line of charge has been bent into an Archimedes Spiral: r(0) = a + (b/Π)θ. Find the electric field at the point x=0, y=0, z=0; as shown in the figure.

Question 2. Let's say that in our lab, we have mapped an electric field with a functional form that looks like this:

F = f(θ, Φ) θ x r where the θ and r are unit vectors in spherical coordinates, f(θ, Φ) is a scalar function that involves both polar variables explicitly.

Show that this can or can not represent a true electrostatic field. Explain what you are doing.

Question 3. Suppose that in a region near the origin there is the electric field

E¯ =(2x² /a², y/a, xy/a²) N/C.

a) Find the flux of this field through the square 0< y

b) What is the net flux of this field through the surface of the cube 0 < x < a, O < y < a, 0 < z < a?

c) Is this electric field conservative?

Question 4. Imagine that new and extraordinarily precise measurements have revealed an error in Coulomb's law. The actual force of interaction between two point charges is found to be

F = 1/4Π∈₀ q₁q₂/r²(1 + r/λ)e^-r/λr,

where λ is a new constant of nature (it has dimensions of length. obviously, and is a huge number-say half the radius of the known universe-so that the correction is small, which is why no one ever noticed the discrepancy before). You are charged with the task of reformulating electrostatics to accommodate the new discovery. Assume the principle of superposition still holds.

(a) What is the electric field of a charge distribution ρ

(b) Does this electric field admit a scalar potential? Explain briefly how you reached your conclusion. (No formal proof necessary-just a persuasive argument.)

(c) Find the potential of a point charge q. (If your answer to (b) was "no," better go back and change it!) Use oo as your reference point.

(d) For a point charge q at the origin, show that

∫_S E.da + 1/λ²∫_vvdτ = 1/∈₀q,

where S is the surface. V the volume, of any sphere centered at q.

(e) Show that this result generalizes:

∫_S E.da + 1/λ²∫_vvdτ = 1/∈₀Q_enc,

for any charge distribution. (This is the next best thing to Gauss's Law, in the new "electrostatics?)

Question 5. A solid conducting sphere of radius 11 is surrounded by a concentric conducting spherical shell of inner radius r7 and outer radius r3. A charge Qi is placed on the inner sphere and a charge Q7 is placed on the outer shell.

a) Find an expression for the electric field as a function of r, the distance to the center of the spheres.

b) Find an expression for the electrostatic potential as a function of r.

c) Make a sketch of the field and potential as a function of r.

d) How would your sketches change if both the sphere and the shell were replaced by uniform volume charge densities giving the same total charge?