A closed loop PQRS carrying a current is placed in a uniform magnetic field. If the magnetic forces on segments PS, SR and RQ are *F*_{1}, *F*_{2} and *F*_{3} respectively and are in the same plane of the paper and along the directions shown, the force on the segment QP is

(1) **√**[(*F*_{3}^{ }– *F*_{1})^{2} + *F*_{2}^{2}]

(2)** √**[(*F*_{3}^{ }– *F*_{1})^{2} – *F*_{2}^{2}]

(3) *F*_{3}^{ }– *F*_{1} + *F*_{2}

(4) *F*_{3}^{ }– *F*_{1} – *F*_{2}

The net force on the loop in the horizontal direction is *F*_{3}^{ }– *F*_{1}. Since the force* F*_{2} is in the vertical direction, the reultant of the three forces *F*_{1}, *F*_{2} and *F*_{3} is **√**[(*F*_{3}^{ }– *F*_{1})^{2} + *F*_{2}^{2}]. Since the net force on the entire loop must be zero in the *uniform* magnetic field, the force on the segment QP must be the equilibrant of **√**[(*F*_{3}^{ }– *F*_{1})^{2} + *F*_{2}^{2}]. Therefore the force on the segment QP has magnitude **√**[(*F*_{3}^{ }– *F*_{1})^{2} + *F*_{2}^{2}] and direction opposite [Option (1)].

Now consider a question which is a little more difficult:

*q*and mass

*m*proceeding along the positive X-direction with speed

*v*encounters a uniform magnetic field of flux density

*B*directed along the negative Z-direction. If the field is confined in the region between

*x*=

*0 and*

*x*=

*d*and the proton emerges from the field along a direction making an angle of 45º with its initial velocity, the value of

*d*must be

(a) (1/√2)(*mv/qB*)* *

(b) √2 *mv/qB*

(c) *mv/qB*

(d) 2* mv/qB*

(e)** ***mv/*2*qB*** **

The situation is shown in the adjoining figure. C is the centre of the circular path of the proton in the magnetic field and AB is the radius *R *drawn from the point A from which the proton exits from the field. We have

*R = mv/qB* which you get by equating the centripetal force *mv*^{2}/*R *to the magnetic force *qvB*.

Since *d = *AN = *R *sin45º where N is the foot of the perpendicular (drawn from A) to the Y-axis, we have

*D = R/*√2 = **(1/√2)( mv/qB)**

You will find many useful questions (with solution) on magnetic force at apphysicsresources