Lorentz force law method

A charge q in the wire on the left side of the loop experiences a Lorentz force q v × B k = −q v B(x_C − w / 2) j   ( j, k unit vectors in the y- and z-directions; see vector cross product), leading to an EMF (work per unit charge) of v ℓ B(x_C − w / 2) along the length of the left side of the loop. On the right side of the loop the same argument shows the EMF to be v ℓ B(x_C + w / 2). The two EMF's oppose each other, both pushing positive charge toward the bottom of the loop. In the case where the B-field increases with increase in x, the force on the right side is largest, and the current will be clockwise: using the right-hand rule, the B-field generated by the current opposes the impressed field.^[13] The EMF driving the current must increase as we move counterclockwise (opposite to the current). Adding the EMF's in a counterclockwise tour of the loop we find