Consider a single rectangular loop PQRS kept in a uniform magnetic field `vecB` . Let a and b be the length and breadth of the rectangular loop respectively . Let ` hatn` be the unit vector normal to the plane of the current loop . This unit vector `hat n` completely describes the orientation of the loop.
Let `vecB` directed from north pole to south pole of the magnet as shown in Figure.
When an electric current is sent through the loop , the net force acting is zero but there will be net torque acting on it. For the sake of understanding, we shall consider two configurations of the loop ,
(i) unit vector `hatn` points perpendicular to the field (ii) unit vector points at an angle `theta ` with the field.
when unit vector `hat n` is perpendicular to the field
In the simple configuration , the unit vector `hatn` is perpendicular to the field and plane of the loop is lying on xy plane as shown in Figure. Let the loop be divided into four sections PQ, QR , RS and SP. The Lorentz force on each loop can be calculated as follows :
(a) Force on section PQ ,
`veci = - a hatj " and " vecB = B hati `
` vecF_("PQ") = vec(Il) xx vecB ( hatj xx hati) = I a B hatk `
Since the unit vector normal to the plane ` hat n` is along the direction of ` hatk `
(b) The force on section QR
`vecl = vec(bi) " and " vecB = B hat i `
` vecF_("QR") = vec(Il) xx vecB = - I bB ( hati xx hati) = vec0 `
( c ) The force on section RS
`vecl = a hatj " and " vecB = B hati `
` vecF_("RS") = vec(Il) xx vecB = I aB ( hatj xx hati) = - I a B hatk `
Since, the unit vector normal to the plane is along the direction of ` - hat k ` .
(d) The force on section SP
`vecl = - b hatj " and " vecB = B hati `
` vecF_("SP") = vec(Il)xx vecB = - IbB ( hati xx hati) = vec0 `
The net force on the rectangular loop is
`vecF_("net") = vecF_("PQ") + vecF_("FS") + vecF_("SP") `
` vecF_("net") = I a B hatk + vec0 - IaB hatk + vecF_("net") = vec0 `
Hence, the net force on the rectangular loop in this configuration is zero . Now let us calculate the net torque due to these dorces about an axis passing through the center
`vectau_("net") = sum_(i=1)^(4) vectau_(i) = sum_(i=1)^(4) vecr_(i) xx vecF_(i) `
` = (b/2 IaB + 0 + b/2 Ia B + 0 ) hatj `
` vectau_("net") = abIB hatj`
Since, A = ab is the area of the rectangular loop PQRS, therefore, the net torque for this configuration is
` vectau_("net") = ABI hatj`
