The positions of two point charges `q_(1)` and `q_(2)` are `vecr_(1)` and `vecr_(2)` , respectively. Find the position of the point where the net field is zero due to thses charges.

Options

A. `(vecr_(1)sqrt(q_(1))+vecr_(2)sqrtq_(2))/(sqrtq_(1)+sqrtq_(2))`
B. `(vecr_(1)sqrt(q_(2))+vecr_(2)sqrtq_(1))/(sqrtq_(1)+sqrtq_(2))`
C. `(vecr_(1)sqrt(q_(2))+vecr_(2)sqrtq_(1))/(sqrt(q_(1)+q_(2)))`
D. `(vecr_(1)sqrt(q_(1))+vecr_(2)sqrtq_(2))/(sqrt(q_(1)+q_(2)))`

Answer

B

Explanation

At P, Let the net field be zero,
`vecE_(P)=vecE_(1)+vecE_(2)=0`…(i)

As we know,
`vecE_(1)=(q_(1)(vecr -vecr_(1)))/(4 pi epsilon_(0)|vecr -vecr_(1)^(3))` …(ii)
and `vecE_(2)=(q_(2)(vecr -vecr_(2)))/(4 pi epsilon_(0)|vecr -vecr_(2)^(3))`
Substituting `vecE_(1)` and `vecE_(2)` from eq.(ii) and (iii)in Eq.(i) we have
`(q_(1)(vec(r )-vec(r_1)))/(4 pi epsilon_(0)|vec(r )-vec(r_1)|^(3))+(q_(2)(vecr -vecr_(2)))/(4 pi epsilon_(0)|vecr -vecr_(2)|^(3))=0`
or, `q_(1)(vecr -vecr_(1))+q_(2)(vec(r )-vecr_(2))*(|vecr -vecr_(1)|^(3))/(|vecr -vecr_(2)|^(3))=0` ...(iv)
Since `E=0` at P, `|vecE_(1)|=|vecE_(2)|`
`implies (q_(1))/(4pi epsilon_(0)r_(P_(1))^(2))=(q_(2))/(4pi epsilon_(0)r_(P_(2))^(2))`
Where `vecr_(P_(1))=vec(r)-vecr_(1)` and `vecr_(P_(2))=vec(r)-vecr_(1)`. Therefore,
`(q_(1))/(|vec(r )-vecr_(1)|^(2))=(q_(2))/(|vec(r )-vecr_(2)|^(2))` or `(|vec(r )-vecr_(1)|)/(|vec(r )-vecr_(2)|)=(sqrtq_(1))/(sqrtq_(2))` ...(v)
Substituting the value of `|vec(r )-vecr_(1)|//|vec(r )-vecr_(1)` from Eq. (v) in Eq. (iv) we get
`q_(1)(vec(r )-vecr_(1))+q_(2)(vec(r )-vecr_(2))((q_(1))/(q_(2)))=0` or `vec(r)=(vecr_(1)sqrt(q_(2))+vecr_(2)sqrtq_(1))/(sqrtq_(1)+sqrtq_(2))`.