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Determine the electric field strength ve...

Determine the electric field strength vector if the potential of this field depends upon x - and y - coordinates as :
(i) `V = a(x^2 - y^2)`
(ii) `V = a x y`.

Text Solution

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(i) As `V = a(x^2 - y^2)`,
`E_x = - (del V)/(del x) = - (del)/(del x)a(x^2 - y^2) = -2 a x`
and `E_y = -(del V)/(del y) = + 2 a y`
`:. vec E = E_x hat i + E_y hat j = - 2 a x hat i + 2a y hat j = - 2 a(x hat i - y hat j)`
(ii) As `V = a x y`,
`E_x = (del V)/(del x) = - a y`
and `E_y = -(del V)/(del y) = - a x`
`:. vec E = E_x hat i + E_y hat j= - a y hat i - a x hat j = - a (y hat i + x hat j)`.
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