Question

A fan blade of length $\dfrac{1}{{\sqrt \pi }}$ meter rotates with frequency 5 cycles per second perpendicular to a magnetic field 10 tesla. What is the potential difference between the centre and the end of the blade?
A. $ - 50V$
B. $ + 50V$
C. $ - 2.0V$
D. $ + 0.02V$

Answer 1

Hint: The potential difference between the two terminals of a cell is called electromotive force. Induced EMF is equal to the rate of change of magnetic flux. Magnetic flux = Magnetic field strength × Area. So a magnetic field is given to find the area covered by the fan. Then substitute the values of area and magnetic field to find the potential difference between centre and end of the blade.

Complete step by step answer:
We are given that A fan blade of length $\dfrac{1}{{\sqrt \pi }}$ meter rotates with frequency 5 cycles per second perpendicular to a magnetic field 10 tesla.
We have to find the potential difference between the centre of the fan and the end of the blade.
Potential difference is a measure of how much energy is transferred between two points in a circuit.
Area covered by the fan in one Time period is $\pi {r^2}$, where the radius will be the length of the fan blade.
$
  r = \dfrac{1}{{\sqrt \pi }} \\
  \pi {r^2} = \pi \times {\left( {\dfrac{1}{{\sqrt \pi }}} \right)^2} \\
  Area = \pi \times \dfrac{1}{\pi } \\
  Area = 1{m^2} \\
 $
Area covered by the fan in one time period is $1{m^2}$
Potential difference (e) =$\dfrac{{ - d\phi }}{{dt}},\phi = B \times A \\ $
$
  e = \dfrac{{ - d\left( {BA} \right)}}{{dt}} \\
  e = \dfrac{{ - BA}}{t} \\
  B = 10,A = 1,f = 5 \\
 $
We know that time period is the reciprocal of frequency.
$
  t = \dfrac{1}{f} \\
  \Rightarrow t = \dfrac{1}{5} \\
  e = \dfrac{{ - BA}}{t} \\
  \Rightarrow e = \dfrac{{ - 10 \times 1}}{{\left( {\dfrac{1}{5}} \right)}} \\
  \Rightarrow e = - 10 \times 5 = - 50V \\
 $
The potential difference between the centre of the fan and the end of fan blade is $ - 50V$
The correct option is Option A, $ - 50V$.

Note:Potential difference is also known as the difference of electrical potential between two points. There is a misconception about potential and voltage. Many of us think that both are the same. But voltage is not exactly potential; it is the measure of the electric potential difference between two points. Voltmeters are used to measure the potential difference between two points.