Question

Find `V_(ab)` in an electric fiels `vec E = (2 hat I + 3 hat j + 4 hat k) NC^-1` where `vec r_a = (hat I - 2 hat j + hat k) m` and `vec r_b = (2 hat i + hat j - 2 hat k) m`.

Answer 1

Here, the given field is uniform (constant). So,
`d V = - vec E . d vec r`
or `V_(ab) = V_a - V_b = - int _b^a vec E . d vec r`
=`-int_((2","1","-2))^((1","-2","1)) (2 hat i + 3 hat j + 4 hat k). (d x hat i + d y hat j + d z hat k)`
=`- int_((2","1","-2))^((1","-2","1))(2 d x + 3 d y + 4 d z)`
=`-[2 x + 3 y + 4 z]_((2","1","-2))^((1","-2","1)) = -1 V`.