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Electromagnetic Induction question

2019 · 10 Jan · Shift 1 · Q45
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  5. /2019 · 10 Jan · Shift 1 · Q45

Electromagnetic Induction question

2019 · 10 Jan · Shift 1 · Q45

JEE MainPhysicsElectromagnetic InductionMCQ+4 / −1
A solid metal cube of edge length 2 cm is moving in a positive y-direction at a constant speed of 6 m/s. There is a uniform magnetic field of 0.1 T in the positive z-direction. The potential difference between the two faces of the cube perpendicular to the x-axis, is -
  1. A
    2mV
  2. B
    12 mV
  3. C
    6 mV
  4. D
    1 mV
View written solutionFree

Correct answer: B

  1. Given data
  • Edge length of cube: a=2 cm=0.02 ma = 2\text{ cm} = 0.02\text{ m}a=2 cm=0.02 m
  • Speed: v=6 m/sv = 6\text{ m/s}v=6 m/s in +y+y+y direction
  • Magnetic field: B=0.1 TB = 0.1\text{ T}B=0.1 T in +z+z+z direction

We need the potential difference between the two faces perpendicular to the xxx-axis.

  1. Direction of magnetic force on charges

For a positive charge,

F⃗=q(v⃗×B⃗)\vec F = q(\vec v \times \vec B)F=q(v×B)

Here,

v⃗∥+j^,B⃗∥+k^\vec v \parallel +\hat j, \qquad \vec B \parallel +\hat kv∥+j^​,B∥+k^

so,

v⃗×B⃗=j^×k^=i^\vec v \times \vec B = \hat j \times \hat k = \hat iv×B=j^​×k^=i^

Thus the magnetic force is along the positive xxx-direction.

So positive charges accumulate on the face at larger xxx, and negative charges on the opposite face. Hence an electric field is established along xxx.

  1. Magnitude of induced electric field

In steady state, electric force balances magnetic force:

qE=qvBqE = qvBqE=qvB

so,

E=vBE = vBE=vB

Substitute values:

E=6×0.1=0.6 V/mE = 6 \times 0.1 = 0.6\text{ V/m}E=6×0.1=0.6 V/m
  1. Potential difference across faces perpendicular to xxx-axis

The separation between those faces is the edge length of the cube:

a=0.02 ma = 0.02\text{ m}a=0.02 m

Therefore,

V=Ea=0.6×0.02=0.012 VV = Ea = 0.6 \times 0.02 = 0.012\text{ V}V=Ea=0.6×0.02=0.012 V

Converting to mV:

0.012 V=12 mV0.012\text{ V} = 12\text{ mV}0.012 V=12 mV
  1. Match with options

The correct option is:

B: 12 mV\boxed{\text{B: }12\text{ mV}}B: 12 mV​
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