Energy gives us the ability to perform physical activities
We lose Energy stored in our body while running and jogging
Energy is dissipated when an object moves from one place to another
To cause the object to move, a Force is applied to it
Work is done by the Force to displace the object from its position
The energy derived from the object due to its Motion is called Kinetic Energy
And the type of energy an object has because of its position is known as the Potential Energy
In other words, an object with potential energy has the potential to do work
Similarly, magnetic materials in a Magnetic field will possess Potential Energy which is related to the orientation, or alignment, of those materials within the Magnetic field
Let us calculate the Potential Energy of a bar magnet placed in a Magnetic Field
The bar magnet has a Magnetic Moment of m→ and it is placed in an Uniform Magnetic field of induction B→
We need to apply Torque to rotate the bar magnet
But to apply Torque on the bar magnet, Work needs to be done on it
We know that Work Done=Torque × Angular Displacement
To rotate the bar magnet by an angle dθ, a Work done of dW needs to be done which can be mathematically expressed as,
dW=τ.dθ
Where,
τ=Torque
dθ=Angular displacement
Torque is mathematically given by,
τ=m→×B→∴τ=mBsinθ
Where m→=Magnetic moment
B→=Magnetic induction
Replacing the value of Torque in the equation,
dW=mBsinθdθ
Now, let's say that originally the magnet is oriented at an angle θ1 with the Magnetic Field
The bar magnet is rotated to a final angle of θ2 with the Magnetic Field
To find the Work Done during this process, we need to integrate dW within θ1 and θ2
Total Work Done W=∫θ1θ2dW
W=∫θ1θ2mBsinθdθ⇒W=mB∫θ1θ2sinθdθ
W=mB[−cosθ]θ1θ2∴W=mB[cosθ1−cosθ2]
It is important to note that the Work done on an object is stored in the form of Potential Energy.
Work done(W)=Potential Energy(U)
Initially the Potential Energy(U)=0 when the angle θ1=90o as cos90o=0
But for an angle θ2=θ the Potential Energy is given by U=−mBcosθ∴U=−m→.B→
Revision
Torque is mathematically given by,
τ=m→×B→∴τ=mBsinθ
Where m→=Magnetic moment
B→=Magnetic induction
Work Done W=mB[cosθ1−cosθ2]
Where,
θ1=Initial angle made by bar magnet with Magnetic field
θ2=Final angle made by bar magnet and Magnetic field after rotation
Work done (W)=Potential Energy (U)
For an angle θ2=θ and θ1=900the Potential Energy is given by U=−mBcosθ∴U=−m→.B→