Question

A square frame of wire abcd of side 1m has a total resistance of 4Ω. It is pulled out of a magnetic field B = 1 T, by applying a force of 1 N. It is found that the frame moves with a constant speed. Find
(a) This constant speed
(b) the emf induced in the loop
(c) the potential difference between a and b
(d) the potential difference between c and d

Answer 1

Hint: Formula for induced electromotive force, magnetic field and current are used. Substituting the values in the formula we can calculate constant speed, electromotive force induced and the voltage across different terminals.
Formula used:
\[\begin{align}
  & F=ILB \\
 & \varepsilon =BvL \\
 & v=\varepsilon -IR \\
\end{align}\]

Complete answer:
A square frame is kept in a magnetic field which is being pulled out by applying a force. After applying force, the frame starts moving with constant speed.
Given-
\[\begin{align}
  & L=1m \\
 & R=4\Omega \\
 & B=1T \\
 & F=1N \\
\end{align}\]
(a)Constant speed
Let the velocity of the frame be, then emf induced in the frame will be
\[\varepsilon =BvL\]
Then we can calculate current in the coil by \[I=\dfrac{BLv}{R}\]……..(1)
Magnetic force in a square frame
\[\begin{align}
  & F=ILB=1N \\
 & I=\dfrac{F}{LB} \\
\end{align}\]……………(2)
Therefore from equation (1) and (2)
\[\dfrac{F}{LB}=\dfrac{BLv}{R}\]

Rearranging the terms
\[v=\dfrac{FR}{{{B}^{2}}{{L}^{2}}}\]
Substituting the given values \[v=\dfrac{1\times 40}{1\times 1}=40m/s\]

(b)Emf induced in the loop
Emf induced in the loop can be calculated by
We know \[\varepsilon =BvL\]
\[=1\times 1\times 40=40V\]

(c)The potential difference between a and b
Total resistance of the frame is \[40\Omega \]
So resistance between a and b \[=\dfrac{40}{4}=10\Omega \]
\[\begin{align}
  & {{V}_{ab}}=\varepsilon -I{{R}_{ab}} \\
 & =40-1(10) \\
 & =30V
\end{align}\]
(d)The potential difference between c and d
Similarly resistance between c and d \[=10\Omega \]
\[\begin{align}
  & {{V}_{cd}}=I{{R}_{cd}} \\
 & =1\times 10=10V
\end{align}\]

Additional information:
Magnetic field is caused due to moving charges. Moving charges create current which results in a magnetic field. In this question the frame is in square shape and current is flowing through it and a magnetic field is produced. Magnetic field has various applications in our daily life.

Note:
Electromotive force is capable of driving electric charge. Resistance across different terminals have been calculated with the help of resistance. Electromotive force always remains constant while the potential difference varies across the terminals.