A cylindrical region of radius R has a time-varying magnetic field B ( r , t ) = B 0 ( 1 − R 2 r 2 ) t k ^ for r ≤ R and B = 0 for r R , where B 0 is a positive constant. Match the quantities

Question

A cylindrical region of radius R has a time-varying magnetic field B ( r , t ) = B 0 ​ ( 1 − R 2 r 2 ​ ) t k ^ for r ≤ R and B = 0 for r > R , where B 0 ​ is a positive constant. Match the quantities in List-I with their magnitudes in List-II. List - I List - II (P) Magnitude of induced electric field at r = R /2 (1) 32 7 B 0 ​ R ​ (Q) Magnitude of induced electric field at r = 2 R (2) 8 B 0 ​ R ​ (R) Magnitude of EMF induced in a circular loop of radius R concentric with the cylindrical region (3) 2 π B 0 ​ R 2 ​ (S) Magnitude of work done by the induced electric field on a charge q moving from ( R , 0 ) to ( 0 , R ) along a circular arc of radius R centered at the origin (4) 8 π q B 0 ​ R 2 ​ (5) 0 (6) 4 B 0 ​ R ​

TestHub · Accepted Answer

For r ≤ R , the induced electric field is E ( r ) = B 0 ​ ( 2 r ​ − 4 R 2 r 3 ​ ) . P: At r = R /2 , E = B 0 ​ ( 4 R ​ − 4 R 2 R 3 /8 ​ ) = 32 7 B 0 ​ R ​ . For r > R , the flux is Φ B ​ ( R ) = 2 π B 0 ​ R 2 t ​ , so E ( r ) = 2 π r 1 ​ ∣ d t d Φ B ​ ( R ) ​ ∣ = 4 r B 0 ​ R 2 ​ . Q: At r = 2 R , E = 4 ( 2 R ) B 0 ​ R 2 ​ = 8 B 0 ​ R ​ . R: The EMF induced in a loop of radius R is E ( R ) ⋅ 2 π R = ( B 0 ​ R /4 ) ⋅ 2 π R = 2 π B 0 ​ R 2 ​ . S: Work done on charge q is W = q ∫ E ⋅ d l = qE ( R ) ⋅ ( arc length ) = q ( B 0 ​ R /4 ) ( π R /2 ) = 8 π q B 0 ​ R 2 ​ .

List - I	List - II
(P) Magnitude of induced electric field at $r = R /2$	(1) $\frac{7 B _{0} R}{32}$
(Q) Magnitude of induced electric field at $r = 2 R$	(2) $\frac{B _{0} R}{8}$
(R) Magnitude of EMF induced in a circular loop of radius $R$ concentric with the cylindrical region	(3) $\frac{π B _{0} R ^{2}}{2}$
(S) Magnitude of work done by the induced electric field on a charge $q$ moving from $(R, 0)$ to $(0, R)$ along a circular arc of radius $R$ centered at the origin	(4) $\frac{π q B _{0} R ^{2}}{8}$
	(5) 0
	(6) $\frac{B _{0} R}{4}$

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