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Question

Consider the cylindrical region of the magnetic field shown in the figure. Region I and II have fields directed perpendicularly outward and inward, respectively. Fields are varying with time as

Region I:B1=3B0t

Region II:B2=B0t

Find the ratio r1r2 providing there is no net induced electric field in the region outside the magnetic field.

A
0.5
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B
0.6
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C
0.3
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D
0.2
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Solution

The correct option is A 0.5
The induced electric field in a closed loop placed in a varying magnetic field is given by,

Edl=dϕBdt

Net magnetic flux,

ϕB=B1πr21B2π(r22r21)

=3B0tπr21B0tπr22+B0tπr21

=4B0tπr21B0tπr22

=B0tπ(4r21r22)

Electric field is zero in the region outside the magnetic field,

dϕBdt=0

B0π(4r21r22)=0

4r21r22=0

r1r2=12=0.5

Hence, option (A) is correct.
Note:


For a region outside the magnetic field i.e. r>R

Edl=dϕdt

E × 2πr=dϕdt

E=dϕdt.12πr

Therefore, electric field even exists (due to the change in flux inside the region) in the region where magnetic field is not present.

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