How to Calculate Electric Field from Rods?

[FS98]

Homework Statement

(a) Two long, thin parallel rods, a distance 2b apart, are joined by a semicircular piece of radius b, as shown in Fig. 1.44. Charge of uniform linear density λ is deposited along the whole filament. Show that the field E of this charge distribution vanishes at the point C. Do this by comparing the contribution of the element at A to that of the element at B which is defined by the same values of θ and dθ.
(b) Consider the analogous two-dimensional setup involving a cylinder and a hemispherical end cap, with uniform surface charge density σ. Using the result from part (a), do you think that the field at the analogous point C is directed upward, downward, or is zero? (No calculations needed!)

Homework Equations

F = kq/d^2

The Attempt at a Solution

I have no idea how to begin this problem. Can somebody explain to me where I can start? I imagine I have to use the equation above and find that the values from each part of the rods cancel out, but I don’t know how without knowing the charge in either case or the distance in the one case.

 

[haruspex]

How long is the shaded element at A? So how much charge on it?
How far is it from C? What is the strength of field it exerts there?

Then the same questions for the other shaded element. For this, it might help to consider the distance of the element from the dashed line, and how that distances changes as the angle increases by dθ.

 

[FS98]

How long is the shaded element at A? So how much charge on it?
How far is it from C? What is the strength of field it exerts there?

Then the same questions for the other shaded element. For this, it might help to consider the distance of the element from the dashed line, and how that distances changes as the angle increases by dθ.

I believe the distance to A is b and the distance to B is b/(cos(θ))

I understand that the charge on each part of the rod is related to the length of that part and that there should be a greater charge at B because of the longer distance. I’m not sure how to calculate these charges though.

Edit: I got a charge of λbdθ for B and λbdθ/cos(θ) for A by multiplying the small angle by the distance. When I try to calculate the electric field at C from these two points, they don’t cancel out. The values I got are kλdθ/b and kcos(θ)λdθ/b. Do you know what I did wrong?

 

[haruspex]

distance to B is b/(cos(θ))

No, I asked for the distance from the dashed line. That is a distance along the straight arm of the wire.
You need the length of the element in that direction. This will be the change in the distance along the arm for a small change dθ in the angle.