Two point charges `Q_(1)` and `Q_(2)` are positioned at points I and 2. The field intensity to the right of the charge `Q_(2)` on the line that passes through the two charges varies according to a law that is represented in the figure. The field intensity is assumed to be positive if its direction coincides with the positive direction on the x-axis. The distance between the charges is l.

(a) Find the sign of each charge
(b) Find the ratio of the absolute values of the charges `|(Q_(1))/(Q_(2))|`
(c) Find the value of b where the field intensity is maximum
[(a) `Q_(2)` is negative and `Q_(1)` is positive (b) `((l +a)/(a))^(2)`, (c) `(l)/(((l+a)/(a))^(2//3)-1)]`

Options

A. `(1)/(Q_(1)//Q_(2)^(1//3)+1)`
B. `(1)/(Q_(1)//Q_(2)^(1//3)-1)`
C. `(1)/(Q_(1)//Q_(1)^(1//3) +1)`
D. `(1)/(Q_(2)//Q_(1)^(1//3)+1)`

Answer

B

Explanation

Electric field at any position to the right of charges
`E=E_1-E_2=1/(4 pi e_0) Q_1/(1+x)^2 -1/(4 pi e_0) Q_2/x^2`
For maximum value of E
`(dE)/(dx)=0 implies d/(dx) [1/(4 pi e_0) Q_1/(1+x)^2-1/(4 pi e_0) Q_2/x^2]=0 , Q_1=(-2)/(1+x)^3-Q_2 (-2)/x^3=0`
`Q_1/(1+x)^3=Q_2/x^3 implies ((1+x)/x)^3 =Q_1/Q_2, (1+x)/x=(Q_1/Q_2)^(1//3) implies 1/x=(Q_1/Q_2)^(1//3) -1 , x=1/((Q_1/Q_2)^(1//3)-1)`