(i) Find the field intensity if potential `V = - k x y (k gt 0)`.
(ii) Draw the field pattern of the field in i.

Explanation

The field intensity
`vec E = - (del V)/(del x) hat i - (del V)/(del y)hat j`
=`- (del)/(del x) (-k x y) hat i (del)/(del y)(- k x y) = k ( y hat i + x hat j)`
(ii) `vec E = E _x hat i + E_y hat j = k y hat i + k y hat j`
or `E_x = k y and E_y = k x`
Then,
`tan theta = (E _y)/(E_x) = (k_x)/(k_y) = (x)/(y)`
Since,
`tan theta = (dy)/(dx)`
(slope of lines of force)
`(d y)/(d x) = (x)/(y)`
or `y d y = x d x`
or `y^2 - x^2 = constant`
Hence, the field pattern is hyperbolic (as shown in Fig. 3.33).
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