Uniform electric field of magnitude `100 Vm^-1` in space is directed along the line `y = 3 + x`. Find the potential difference between points `A (3,1) and B (1,3)`.

Explanation

Equation of the line
`y = 3 + x`.
The slope of line
`tan theta = 1 or theta = 45^(@)`
we can express electric field in vector form
`vec E = 100 cos theta hat i + 100 sin theta hat j (V m^-1)`
Electric field `vec E = |vec E | (cos theta hat i + sin theta hat j)`
`vec E = (100)/(sqrt(2)) hat i + (100)/(sqrt(2)) hat j (V m^-1)`
`Delta vec r = vec (AB) = vec r_B - vec r _A = (hat i + 3 hat j)-(3 hat i + hat j)= -2 hat i + 2 hat j`
`Delta V = - vec E . Delta vec r = - ((100)/(sqrt(2)) hat i + (100)/(sqrt(2)) hat j). Delta vec r`
= `- ((100)/(sqrt(2)) hat i + (100)/(sqrt(2)) hat j) . (-2 hat i + 2 hat j)`
=`-100 sqrt(2) + 100 sqrt(2)`
`V_A - V_B = 0`.