A coil of turns has its plane initially perpendicular to a uniform magnetic field

PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences

Problem

A coil of

N

turns has its plane initially perpendicular to a uniform magnetic field. The quantities that may vary with time are given below: • Magnetic field:

B(t)=B_0(1-\beta t)

where

B_0

and

\beta

are positive constants. • Area of each turn:

A(t)=A_0\left(1+\alpha t^{2}\right)

where

A_0

and

\alpha

are positive constants. • The coil is simultaneously rotated with a uniform angular speed

\omega

about an axis in its own plane so that the angle between its normal and the field at any instant is

\theta(t)=\omega t

. Assume

t\ge 0

and that all quantities vary smoothly. (i) Derive an explicit expression for the instantaneous induced emf

\mathcal{E}(t)

in the coil in terms of

N,\;B_0,\;A_0,\;\alpha,\;\beta,\;\omega,

and

t

. (ii) Using your result from part (i), evaluate

\mathcal{E}

t=1.0\,\text{s}

for the following numerical values:

N=200

B_0=0.80\,\text{T}

\beta=0.10\,\text{s}^{-1}

A_0=8.0\times10^{-4}\,\text{m}^2

\alpha=0.50\,\text{s}^{-2}

and

\omega=10\,\text{rad s}^{-1}

. Express the answer correct to two significant figures. (iii) Comment on which of the three time-dependent processes (change in

B

, change in

A

, or rotation) makes the largest contribution to the induced emf at

t=1.0\,\text{s}

for the given data. Give a brief quantitative justification.