A planar loops of wire rotates in a uniform magnetic field. Initially, at $t = 0$ , the plane of the loop is perpendicular to the magnetic field. If it rotates with a period of $10 s$ about an axis in its plane then the magnitude of induced emf will be maximum and minimum, respectively at :

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Answer 1

Time period, $T = 10 s$
Ref. image I

Axis of rotation $\to$ y-axis.

$B = B (- \hat{k})$

ref. image II
at some time t, area vector makes angle ' $θ$ ' with $B$ .
flux, $ϕ = B . A$

$= B A cos θ$

emf, $ε = ∣ ∣ ∣ ∣ ∣ \frac{- d ϕ}{d t} ∣ ∣ ∣ ∣ ∣ = (B A sin θ \frac{d θ}{d t})$

$= B A ω sin θ$

When $θ = \frac{π}{2}, ε = ε_{m a x}$

So, $θ = n π + \frac{π}{2}$ n = integes

When $θ = 0, ε = ε_{m i n} = 0$

So, $θ = n π$ , n = integes

for $ε = ε_{m a x}, θ = ω t = n π + \frac{π}{2}$

$\Rightarrow t = \frac{n π}{ω} + \frac{π}{2 ω} = \frac{n π}{( \frac{2 π}{T} )} + \frac{π}{2 \times ( \frac{2 π}{T} )}$

$t = ⎝ ⎜ ⎜ ⎛ \frac{n π}{\frac{2 π}{T}} ⎠ ⎟ ⎟ ⎞ + \frac{π}{2 ( \frac{2 π}{T} )}$

$= \frac{n T}{2} + \frac{T}{4} = \frac{n \times 1 0}{2} + \frac{1 0}{4}$
$= 5 n + 2.5$

$= 2.5 s, 5 + 2.5, 5 \times 2 + 2.5 . . .$

$= 2.5 s, 7.5 s, 12.5 s . . . .$

for $ε = ε_{m i n}, θ = ω t = n π$

$\Rightarrow t = \frac{n π}{ω} = \frac{n π}{( \frac{2 π}{T} )} = \frac{n T}{2}$

$\Rightarrow t = \frac{n \times 1 0}{2} = 5 n$

$= 0 s, 5 s, 10 s, . . .$

So, possible option is option (C).

Answer 2

Time period,

T = 10 s

Ref. image I

Axis of rotation

\to

y-axis.

B = B (- \hat{k})

ref. image II

at some time t, area vector makes angle '

θ

' with

B

.

flux,

ϕ = B . A

= B A cos θ

emf,

ε = ∣ ∣ ∣ ∣ ∣ \frac{- d ϕ}{d t} ∣ ∣ ∣ ∣ ∣ = (B A sin θ \frac{d θ}{d t})

= B A ω sin θ

When

θ = \frac{π}{2}, ε = ε_{m a x}

So,

θ = n π + \frac{π}{2}

n = integes

When

θ = 0, ε = ε_{m i n} = 0

So,

θ = n π

, n = integes

for

ε = ε_{m a x}, θ = ω t = n π + \frac{π}{2}

\Rightarrow t = \frac{n π}{ω} + \frac{π}{2 ω} = \frac{n π}{( \frac{2 π}{T} )} + \frac{π}{2 \times ( \frac{2 π}{T} )}

t = ⎝ ⎜ ⎜ ⎛ \frac{n π}{\frac{2 π}{T}} ⎠ ⎟ ⎟ ⎞ + \frac{π}{2 ( \frac{2 π}{T} )}

= \frac{n T}{2} + \frac{T}{4} = \frac{n \times 1 0}{2} + \frac{1 0}{4}

= 5 n + 2.5

= 2.5 s, 5 + 2.5, 5 \times 2 + 2.5 . . .

= 2.5 s, 7.5 s, 12.5 s . . . .

for

ε = ε_{m i n}, θ = ω t = n π

\Rightarrow t = \frac{n π}{ω} = \frac{n π}{( \frac{2 π}{T} )} = \frac{n T}{2}

\Rightarrow t = \frac{n \times 1 0}{2} = 5 n

= 0 s, 5 s, 10 s, . . .

So, possible option is option (C).

A planar loops of wire rotates in a uniform magnetic field. Initially, at $t = 0$ , the plane of the loop is perpendicular to the magnetic field. If it rotates with a period of $10 s$ about an axis in its plane then the magnitude of induced emf will be maximum and minimum, respectively at :

$5.0 s$ and $7.5 s$

$2.5 s$ and $7.5 s$

$2.5 s$ and $5.0 s$

$5.0 s$ and $10.0 s$

In a given figure an isosceles triangle conducting wire is entering into the region of perpendicular uniform magnetic field $B_{0}$ with constant velocity $V_{0}$ . The magnitude of induced emf as a function of time(t) in the wire is(till side is outside the field)

A conducting ring of radius $1$ metre is placed in a uniform magnetic field $B$ of $0.01$ tesla oscillating with frequency $100 H z$ with its plane at right angle to $B$ . What will be the maximum induced electric field?

The phase difference between induced emf and magnetic flux linked with a coil rotating in a uniform magnetic field is:

Revise with Concepts

Gauss Law for Magnetism

Faraday Law Lenz's Law - Problem L1

A planar loops of wire rotates in a uniform magnetic field. Initially, at t=0, the plane of the loop is perpendicular to the magnetic field. If it rotates with a period of 10s about an axis in its plane then the magnitude of induced emf will be maximum and minimum, respectively at :

5.0s and 7.5s

2.5s and 7.5s

2.5s and 5.0s

5.0s and 10.0s

In a given figure an isosceles triangle conducting wire is entering into the region of perpendicular uniform magnetic field B0​ with constant velocity V0​. The magnitude of induced emf as a function of time(t) in the wire is(till side is outside the field)

A conducting ring of radius 1 metre is placed in a uniform magnetic field B of 0.01 tesla oscillating with frequency 100Hz with its plane at right angle to B. What will be the maximum induced electric field?

The phase difference between induced emf and magnetic flux linked with a coil rotating in a uniform magnetic field is:

Revise with Concepts

Gauss Law for Magnetism

Faraday Law Lenz's Law - Problem L1

A planar loops of wire rotates in a uniform magnetic field. Initially, at $t = 0$ , the plane of the loop is perpendicular to the magnetic field. If it rotates with a period of $10 s$ about an axis in its plane then the magnitude of induced emf will be maximum and minimum, respectively at :

$5.0 s$ and $7.5 s$

$2.5 s$ and $7.5 s$

$2.5 s$ and $5.0 s$

$5.0 s$ and $10.0 s$

In a given figure an isosceles triangle conducting wire is entering into the region of perpendicular uniform magnetic field $B_{0}$ with constant velocity $V_{0}$ . The magnitude of induced emf as a function of time(t) in the wire is(till side is outside the field)

A conducting ring of radius $1$ metre is placed in a uniform magnetic field $B$ of $0.01$ tesla oscillating with frequency $100 H z$ with its plane at right angle to $B$ . What will be the maximum induced electric field?