Learn formula of average induced EMF and magnetic flux, calculate average induced EMF inside the solenoid. Practice to find the magnitude of average induced EMF in that period of time.
Case 1 When bar magnet is stationary
The magnetic flux linked with coil will be constant as the number of field lines passing through coil is constant.
Case 2 When bar magnet is moved closer to coil
The magnetic flux linked with coil will increase as the number of field lines passing through coil is increasing.
Case 3 When bar magnet is moved away from coil
The magnetic flux linked with coil will decrease as the number of field lines passing through coil is decreasing.
This current is known as induced current and it is produced by induced emf.
\(\mathcal{E} = \dfrac{- d \phi_m}{dt}\)
\(\mathcal{E} = \dfrac{- d }{dt}(BA\,cos\theta)\) [\(\because\) \(\phi_m = BA \,cos\theta\)]
\(\mathcal{E} = -N \dfrac{d\,\phi_m}{dt}\)
\(\phi _m = \vec B \cdot \vec A\)
\(\left|\dfrac{d}{dt}\phi_m \right| = \left|\vec B \cdot \dfrac{d \vec A}{dt} + \vec A \cdot \dfrac{d \vec B}{dt}\right|\)
\(\mathcal E = \left| \vec B \cdot \dfrac{d\,\vec A}{dt} + \vec A \cdot \dfrac{d \vec B}{dt} \right| \)
i) When area vector \(\vec A\) is constant and magnetic field \(\vec B\) varies
\(\mathcal E =\vec A \cdot \dfrac{d\vec B}{dt}\)
\(\mathcal E_{avg} =\vec A \cdot \dfrac{\Delta\vec B}{\Delta t}\)
ii) When magnetic field is constant and area vector varies
\(\mathcal E_{avg} =\vec B \cdot \dfrac{\Delta\vec A}{\Delta t}\)
\(B = \mu_0\,nI\)
\(\dfrac{\Delta B}{\Delta t} = \mu_0 \,n \dfrac{\Delta I}{\Delta t}\)
A \(4 \pi^2 × 10^{-8 }\ V\)
B \(2 \pi^2 × 10^{-6 }\ V\)
C \(8 \pi^2 × 10^{-3 }\ V\)
D \(2 \pi^2 × 10^{-4 }\ V\)
Consider two coaxial solenoids such that solenoid of radius \(r_1\) is placed inside a solenoid of radius \(r_2\), coaxially.
The number of turns in inner and outer solenoid is \(N_1 \,\,\text{and }\,\, N_2\), respectively.
Case 1:
\(A= \pi\,r_1^2 × \text{number of turns}\)
\(= N_1 \,\pi\,r_1^2\)
\(B=\mu_0 N_2I\)
\(\phi=B.A\)
\(\dfrac{\Delta \phi}{\Delta t} = \dfrac{\Delta (B.A)}{\Delta t}\)
\(\dfrac{\Delta \phi}{\Delta t} = A\dfrac{\Delta B}{\Delta t}\)
\(\dfrac{\Delta \phi}{\Delta t} = N_1\,\pi\,r_1^2 . \mu_0\,N_2\,\dfrac{\Delta \,I}{\Delta t}\)
Case 2:
\(A= \pi\,r_1^2 × \text{number of turns}\)
\(A= \pi\,r_1^2× N_2\)
\(B= \mu_0\,N_1 \,I\)
\(\phi =B.A\)
\(\dfrac{\Delta \phi}{\Delta t} =\dfrac{\Delta }{\Delta t}(B.A) \)
\(\dfrac{\Delta \phi}{\Delta t} =A\left(\dfrac{\Delta B }{\Delta t}\right)\)
\(\dfrac{\Delta \phi}{\Delta t} =\pi\,r_1^2 \,N_2 \,\,\mu_0\,N_1\dfrac{\Delta I }{\Delta t}\)
A \(0\cdot8 \pi^2 \,Wb\)
B \(0\cdot4 \pi^2 \,Wb\)
C \(0\cdot6 \pi^2 \,Wb\)
D \(0\cdot9 \pi^2 \,Wb\)
The magnetic field and area remains constant but due to flipping , the magnetic flux through the coil changes as the angle between the area vector and magnetic field changes.
At initial position
Magnetic flux is
\((\phi_m)_i = BA\,cos 0^\circ\)
\((\phi_m)_i = BA\)
At final position
Magnetic flux is
\((\phi_m)_f = BA\,cos 180°\)
\((\phi_m)_f = -B\,A\)
\(\Delta \phi_m = |(\phi_m)_f-(\phi_m)_i|\)
\(\Delta \phi_m = |-BA-BA|\)
\(\Delta \phi_m = 2\,BA\)
Then average induced emf,
\(\mathcal E_{avg} = \left|\dfrac{\Delta \phi}{\Delta t}\right|\)
\(\mathcal E_{avg} = \left|\dfrac{2BA}{\Delta t}\right|\)
Note:
\(\phi _m = \vec B \cdot \vec A\)
\(\left|\dfrac{d}{dt}\phi_m \right| = \left|\vec B \cdot \dfrac{d \vec A}{dt} + \vec A \cdot \dfrac{d \vec B}{dt}\right|\)
\(\mathcal E= \left| \vec B \cdot \dfrac{d\,\vec A}{dt} + \vec A \cdot \dfrac{d \vec B}{dt} \right| \)
i) When area vector \(\vec A\) is constant and magnetic field \(\vec B\) varies
\(\mathcal E =\vec A \cdot \dfrac{d\vec B}{dt}\)
\(\mathcal E_{avg} =\vec A \cdot \dfrac{d\vec B}{dt}\)
ii) When magnetic field is constant and area vector varies
\(\mathcal E_{avg} =\vec B \cdot \dfrac{d\vec A}{dt}\)
Average induced emf is given as
\(\left|\mathcal E_{avg}\right| = \dfrac{\Delta\phi}{\Delta t}\)
\(\left|\mathcal E_{avg}\right| = \dfrac{\Delta(BA)}{\Delta t}\)
\(\left|\mathcal E_{avg}\right| = B\left(\dfrac{\Delta A}{\Delta t}\right)\)
\(\left|\mathcal E_{avg}\right| = \dfrac{B(A_f - A_i)}{\Delta t}\)