Question

Two point charges + Q, and-Q, are placed r distance apart. Obtain
the expression for the amount of work done to place a third charge
Q3 at the midpoint of the line joining the two charges.
(b) At what distance from charge + Q, on the line joining the two
charges (in terms of Q1, Q2 and r) will this work done be zero.​

Answer 1

Answer:

(a)as we know:W=q*V

now work done to place the third charge charge at mid point:

let the distance between two charges +Q and -Q be "r" ,

now their mid point is r/2

therefore,W=q(Q3)*V{(+2Q/4$\pi$∈r)+(-2Q/4$\pi$∈r)}

W=Q3*2/4$\pi$∈r(+Q-Q)

W=Q3*2/4$\pi$∈r(0)

W=0

(b)we have already found out were W is 0.but still.......

let at a point P at a distance x from +Q ,V is zero hence W is also zero at  

distance x because W=q*V

if V =0,then W=0

now,V= (+Q/4$\pi$∈x)+(-Q/4$\pi$∈(r-x))

A.T.Q :(+Q/4$\pi$∈x)+(-Q/4$\pi$∈(r-x))=0

(1/x-1/(r-x))*Q/4$\pi$∈=0

(1/x-1/(r-x)=0

1/x=1/(r-x)

x=r-x

2x=r

therefore,x=r/2

hence at r/2 distance from +Q ,W is zero.

Explanation:(a) Here q=Q3 (charge that we want to place )

V=(potential difference due to charge +Q at distance r/2)+(potential difference due to charge +Q at distance r/2)

(b)x=distance between +Q and point P

(r-x)=distance between -Q and point P

Answer 2

The expression for the work done W = (Q₃/2πε₀r)(Q₁ - Q₂)

and the required distance is Q₁r/ (Q₂ + Q₁).  

Given:

Two point charges + Q₁, and - Q₂, are placed r distance apart and the third charge is Q₃ at the midpoint of the line joining the two charges.

To find:

(a) Obtain the expression for the amount of work done to place a third charge Q₃  

(b) At what distance from charge + Q, on the line joining the two charges (in terms of Q₁, Q₂, and r) will this work done be zero?​  

Solution:

Let A, B be the point at charges + Q₁, and - Q₂ and C be the point at Q₃

To find the work done find the potential difference between infinity to point C i.e that is given by U$_{c}$ - U$_{\infty}$ Hence

Work done,  W = Q₃(U$_{c}$ - U$_{\infty}$)  

=> W = Q₃(U$_{c}$)             [ ∵ U$_{\infty}$ = 0 ]    

As we know potential at Q₃ is the potential difference between Q₁ and Q₂  

=> W = Q₃ [ Q₁/4πε₀(r/₂) - Q₂/4πε₀(r/₂) ]

=> W = Q₃ [ Q₁/2πε₀r - Q₂/2πε₀r]  

=> W = Q₃ (1/2πε₀r)(Q₁ - Q₂)  

=> W = (Q₃/2πε₀r)(Q₁ - Q₂)      

Hence, The work done W = (Q₃/2πε₀r)(Q₁ - Q₂)    

Here, when the potential of point C becomes zero then the work done, will be zero. Let 'x' be the distance of point c from Q₁ and the potential is zero  

Hence, the net potential at C

=>  K [ Q₁/x - Q₂/(r-x) ] = 0

=>  [ Q₁/x - Q₂/(r-x) ] = 0

=>  [ Q₁/x - Q₂/(r-x) ] = 0  

=>  Q₁/x = Q₂/(r-x)  

=>  Q₁r- Q₁x = Q₂x  

=>  Q₁r = Q₂x + Q₁x

=> x(Q₂ + Q₁) = Q₁r

=> x = Q₁r/ (Q₂ + Q₁)        

The expression for the work done W = (Q₃/2πε₀r)(Q₁ - Q₂)

and the required distance is Q₁r/ (Q₂ + Q₁)  

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