Question

The electric field strength depends only on the x and y coordinates according to the law E = (a(xhati+ y hatj))/(x^(2) + y^(2)) , where a is a constant hati and hatj are unit vectors of the x and y axes . Find the potential difference between x = 1 and x = 5 :

Answer 1

Hence the potential difference between the points is V = - a ln 5

Explanation:

The given points are:

( 1 , 0 ) and ( 5 , 0 )

V = ∫ P2 - P1  E.dl

E = a ( x i + y j ) / x^2 + y^2

dl = d xi + d yj

V = ∫P2 - P1  ax  dx / x^2 + y^2 + ay dy / x^2 + y^2

[ Since  x^2 + y^2 = t ]

[ 2 x = dt / dx ]

[ 2x dx = dt ]

a =  [ 1 / 2 ∫ P2 - P1  2x dx /  x^2 + y^2

a = 1 / 2  ∫ P2 - P1  dt / t

[ Since  ∫ dt / t = ln t ]

a / 2 lnt ] P2 - P1 => a / 2 ln (x^2 + y^2 )] ^(5,0) - (1 ,0)

- a / 2 ln 5 ^2

V = - a ln 5

Hence the potential difference between the points is V = - a ln 5