A conducting rod with zero resistance and length w=0.2 m slides without friction

Question

A conducting rod with zero resistance and length w=0.2 m slides without friction on two parallel and perfectly conducting wires. Resistors R1=350 Ohm and R2=700 Ohm are connected across the ends of the wires to form a circuit, as shown. A constant and uniform magnetic field |B^rightarrow| = 3 T is directed out of the page. In computing the magnetic flux through any surface, take the surface normal to be out of the page, parallel to B. Is the magnetic flux in the right loop of the circuit decreasing or increasing? Is the magnetic flux in the left loop of the circuit decreasing or increasing? What is the magnitude of the EMF through the right loop? What is the magnitude of the EMF through the left loop? What is the current flowing through the resistor R1 in the left hand loop of the circuit shown? Gives its magnitude and indicate its direction (CW or CCW). What is the current flowing through the resistor R2 in the right hand loop of the circuit shown? Give its magnitude and indicate its direction (CW or CCW). What is the magnitude and direction of the magnetic force exerted on this rod?

Explanation / Answer

part a:

magnetic flux=mangetic field *area

as the rod is moving to the left, area of the right region is increasing

hence magnetix flux is also increasing.

part b:

area of the left loop is decreasing

hence magnetic flux is also decreasing.

part c:

magnitude of emf induced =magnitude of rate of change of flux linkage

=rate of change of (B*area)

=B*rate of change of area

=B*w*v

=3*0.2*v

=0.6*v

so magnitude of emf induced in the right loop is 0.6*v.

part d:

magnitude of emf induced in left loop is same as right loop and equal to 0.6*v.

part e:

current through R1=emf induced in left loop/resistance of R1

=0.6*v/350 A

=1.7143*v mA

as flux is decreasing in the left loop, as per lenz's law, induced current be such that it will oppose the original flux.

hence current is in CW direction.

part f:

current through R2=emf induced in left loop/resistance of R2

=0.6*v/700 A

=0.85714*v mA

as flux is increasing in the right loop, as per lenz's law, induced current be such that it will oppose the original flux.

hence current is in CCW direction.

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