A long cylindrical region of radius R has a uniform magnetic field B = B 0 sin ( ω t ) k ^ for r ≤ R and B = 0 for r R , where k ^ is along the cylinder axis. Match the quantities in List-I with t

Question

A long cylindrical region of radius R has a uniform magnetic field B = B 0 ​ sin ( ω t ) k ^ for r ≤ R and B = 0 for r > R , where k ^ is along the cylinder axis. Match the quantities in List-I with their corresponding expressions in List-II. List - I List - II (P) Magnitude of the induced electric field at a point r = R /2 (1) B 0 ​ ω R 2 ∣ cos ( ω t ) ∣ (Q) Magnitude of the induced electric field at a point r = 2 R (2) 4 B 0 ​ ω R ​ ∣ cos ( ω t ) ∣ (R) Magnitude of the EMF induced in a square loop of side L = R placed concentrically with its sides parallel to x and y axes, entirely within r ≤ R (3) − q B 0 ​ ωπ R 2 cos ( ω t ) (S) Work done by the induced electric field in moving a charge q once around a circular path of radius r 0 ​ = R (concentric with cylinder) (4) 2 r B 0 ​ ω R 2 ​ ∣ cos ( ω t ) ∣ (5) 2 B 0 ​ ω R ​ ∣ cos ( ω t ) ∣ (6) 2 B 0 ​ ω r ​ ∣ cos ( ω t ) ∣

TestHub · Accepted Answer

From Faraday's law, ∮ E ⋅ d l = − d t d Φ B ​ ​ . For r ≤ R , Φ B ​ = B 0 ​ sin ( ω t ) π r 2 , so E ( 2 π r ) = − B 0 ​ ω cos ( ω t ) π r 2 , giving E = − 2 B 0 ​ ω r ​ cos ( ω t ) . For r > R , Φ B ​ = B 0 ​ sin ( ω t ) π R 2 , so E ( 2 π r ) = − B 0 ​ ω cos ( ω t ) π R 2 , giving E = − 2 r B 0 ​ ω R 2 ​ cos ( ω t ) . For (P) at r = R /2 : ∣ E ∣ = 2 B 0 ​ ω ( R /2 ) ​ ∣ cos ( ω t ) ∣ = 4 B 0 ​ ω R ​ ∣ cos ( ω t ) ∣ . So (P)-(2). For (Q) at r = 2 R : ∣ E ∣ = 2 ( 2 R ) B 0 ​ ω R 2 ​ ∣ cos ( ω t ) ∣ = 4 B 0 ​ ω R ​ ∣ cos ( ω t ) ∣ . So (Q)-(2). For (R), the flux through a square loop of side R (area R 2 ) within r ≤ R is Φ B ​ = B 0 ​ sin ( ω t ) R 2 . The induced EMF magnitude is ∣ − d t d Φ B ​ ​ ∣ = ∣ − B 0 ​ ω R 2 cos ( ω t ) ∣ = B 0 ​ ω R 2 ∣ cos ( ω t ) ∣ . So (R)-(1). For (S), work done W = q ∮ E ⋅ d l . For a circular path of radius r 0 ​ = R , ∮ E ⋅ d l = − d t d ​ ( B 0 ​ sin ( ω t ) π R 2 ) = − B 0 ​ ωπ R 2 cos ( ω t ) . Thus, W = − q B 0 ​ ωπ R 2 cos ( ω t ) . So (S)-(3).

List - I	List - II
(P) Magnitude of the induced electric field at a point $r = R /2$	(1) $B_{0} ω R^{2} ∣ cos (ω t) ∣$
(Q) Magnitude of the induced electric field at a point $r = 2 R$	(2) $\frac{B _{0} ω R}{4} ∣ cos (ω t) ∣$
(R) Magnitude of the EMF induced in a square loop of side $L = R$ placed concentrically with its sides parallel to x and y axes, entirely within $r \leq R$	(3) $- q B_{0} ωπ R^{2} cos (ω t)$
(S) Work done by the induced electric field in moving a charge $q$ once around a circular path of radius $r_{0} = R$ (concentric with cylinder)	(4) $\frac{B _{0} ω R ^{2}}{2 r} ∣ cos (ω t) ∣$
	(5) $\frac{B _{0} ω R}{2} ∣ cos (ω t) ∣$
	(6) $\frac{B _{0} ω r}{2} ∣ cos (ω t) ∣$

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JEE Advanced 2027 (Class 12)