I. Three point charges are located along a line as shown below. Their charges ar

Question

I. Three point charges are located along a line as shown below. Their charges are: qa = +1 pC, qb = +Luc, and qc--1 pC (a) Find the location along the line between charge a and charge b where the net electric field due to all 3 charges is zero; give your answer as the distance of the point from charge b, to 3 significant figures. You are not allowed to use a computer or calculator to do any of the algebra involved, nor to solve the guartic equation that results; you may use the quadratic formula, and a calculator to evaluate a quadratic formula. Note: I strongly recommend you read the hint for this part, below. (b) Is there any point along the line to the left of charge a where the net electric field is zero? Explain. (c) Is there any point along the line between charge b and charge c where the net electric field is zero? Explain. LOm 1.0 m

Explanation / Answer

a) Let q = qa = qb = -qc = 1 * 10-6 C

Let the point b 'r' to the left of qb. Also, the direction to the right be positive. Then,

Ea = kq/(1 - r)2

Eb = -kq/r2

Ec = kq/(1 + r)2

Now, Enet = Ea + Eb + Ec = 0

=> kq[1/(1 - r)2 - 1/r2 + 1/(1 + r)2] = 0

=> 1/r2 = 1/(1 - r)2 + 1/(1 + r)2

=> 1/r2 = (2r2 + 2) / (1 - r2)2

=> 1 + r4 - 2r2 = 2r4 + 2r2

Let x = r2. Then,

1 + x2 - 2x = 2x2 + 2x

=> x2 + 4x - 1 = 0

=> x = {-4 + [42 - (4 * 1 * (-1))]1/2} / (2 * 1)

=> r = x1/2 = [ {-4 + [42 - (4 * 1 * (-1))]1/2} / (2 * 1) ]1/2 = 0.486 m

b) Electric field at points left of 'a' due to charges 'a' and 'b' is directed left whereas the electric field due to charge 'c' is towards right.

Since the strengths of the charges is the same and the points left of charge qa are more close to 'a' and 'b' than 'c', net electric field cannot be zero.

c) At points between charges 'b' and 'c', the electric field due to all charges is towards right, so the net electric field cannot be zero there.

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