Question

Figure shows a rectangular loop conducting PQRS in which the arm PQ is free to move. A uniform magnetic field acts in the direction perpendicular to the plane of the loop. Arm PQ is moved with the velocity v towards the arm RS. Assuming that the arms QR,RS and SP have negligible resistance and the moving arm PQ has the resistance r, obtain the expression for i) current ii) force iii) and the power required to move the arm PQ

Answer 1

Hint: In this question, we will use the relation between current and e.m.f to find the current, force and current to find the force and the last power and force to find the power. These equations will give us the required expression. Further, we will study the basics of power, force, speed and velocity as well, for our better understanding.
Formula used:
$I = \dfrac{\varepsilon }{r}$
$F = BIl$
$P = Fv$
Complete step by step solution: as we know that an electric current is defined as the stream of charged particles, such as electrons or ions, moving through any electrical conductor or space.
Current in the loop PQRS will be given as:
$I = \dfrac{\varepsilon }{r}$
As we know that e.m.f is defined as the rate of change of flux with time:
$\varepsilon = \dfrac{{d\phi }}{{dt}} = Blv$
Now, by substituting the values in the current equation, we get:
$ \Rightarrow I = \dfrac{{Blv}}{r}$
ii) Now, when we calculate the force, we have:
$F = BIl$
Substituting the value of current in the above equation and solving, we get:
$\eqalign{& \Rightarrow F = B\left( {\dfrac{{Blv}}{r}} \right)l \cr
  & \Rightarrow F = \dfrac{{{B^2}{l^2}v}}{r} \cr} $
iii) Now, to find the power required to move the arm PQ we will use:
$P = Fv$
By substituting the value of force in above equation and solving, we get:
$ \Rightarrow P = \left( {\dfrac{{{B^2}{l^2}v}}{r}} \right)v$
$\therefore P = \left( {\dfrac{{{B^2}{l^2}{v^2}}}{r}} \right)$
Therefore we get the required answer, which gives us the value of current, force and power.

Additional information:
We should also know about the power. It is the amount of energy transferred or converted per unit time. In the International System or S.I Unit of power is the watt. The S.I unit of power can also be written as one joule per second. Power is a scalar quantity, which means it has only magnitude not direction.
Force is simply any push or pull. The S.I unit of force is Newton represented by N. The acceleration is defined as the increase in the velocity of an object. The acceleration is measured in meters per Second Square. There are three laws of motions given by Newton. These laws of motion relate an object's motion to the forces acting on it.
First law of motion states that an object continues to be in rest or in motion in a particular direction until and unless any external force is applied on it.
 In the second law of motion, the force on an object is equal to its mass times its acceleration. This law also gives the relation of momentum and force.
In the third law of motion, every action has an equal and opposite reaction.
We already know the basic difference of speed and velocity i.e., speed is the measure of how fast an object can travel, whereas velocity tells us the direction of this speed. Speed is a scalar quantity that means it has only magnitude, whereas velocity is a vector quantity that means it has both magnitude and direction. The S.I unit of velocity is meter per second (m/sec).

Note:
Force applied on an object changes its motion, speed, direction and also its shape. There is a very small difference between the force applied and the pressure applied on an object. Pressure is always given as the force per unit area. Power is measured in watt.