Question

(a) Given two point charges, Q and 2Q, a distance d apart, is there a point along the straight line that passes through them where E = 0 when their signs are opposite? If yes, state roughly where this point will be.
(b) Given two point charges, Q and 2Q, a distance d apart, is there a point along the straight line that passes through them where E = 0 when their signs are the same? If yes, state roughly where this point will be.

Answer 1

Part A. Electric field is given by:

E = kq/R^2

Since we know that direction of electric field due to +ve charge will be away from charge and due to -ve charge will be towards the charge. In given case since q1 and q2 have opposite sign, So electric field cannot be zero between charges and It will be zero towards the left side of q1 (Since |q2| > |q1|)

Assuming q1 = Q, q2 = 2*Q, and suppose q1 is at origin and q2 is at x = +d

Now suppose at point P, distance x = -r electric field is zero, then

Enet = E1 - E2 = 0

E1 = E2

kq1/r1^2 = kq2/r2^2

r1 = distance between q1 and P = r

r2 = distance between q2 and P = d + r

Using above values:

q1/r1^2 = q2/r2^2

Q/r^2 = (2Q)/(d + r)^2

1/r^2 = 2/(d + r)^2

(d + r)/r = sqrt 2

d + r = r*sqrt 2

r = d/((sqrt 2) - 1) = 2.414*d

So electric field will be zero at '2.414*d' distance from weaker charge Q, and at '3.414*d' distance from the stronger charge 2Q.

Part B. Electric field is given by:

E = kq/R^2

Since we know that direction of electric field due to +ve charge will be away from charge and due to -ve charge will be towards the charge. In given case since q1 and q2 have same sign, So electric field will be zero between charges.

Assuming q1 = Q, q2 = 2*Q, and suppose q1 is at origin and q2 is at x = +d

Now suppose at point P, distance x = r from origin electric field is zero, then

Enet = E1 - E2 = 0

E1 = E2

kq1/r1^2 = kq2/r2^2

r1 = distance between q1 and P = r

r2 = distance between q2 and P = d - r

Using above values:

q1/r1^2 = q2/r2^2

Q/r^2 = (2Q)/(d - r)^2

1/r^2 = 2/(d - r)^2

(d - r)/r = sqrt 2

d - r = r*sqrt 2

r = d/((sqrt 2) + 1) = 0.414*d

So electric field will be zero at '0.414*d' distance from weaker charge Q, and at '0.586*d' distance from the stronger charge 2Q.

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