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Home > News & Articles > Magnetic Induction Formula: Faraday's Law of Induction, Application and Sample Questions
Updated on 03rd May, 2023 , 7 min read
Magnetic induction is a phenomenon that occurs when a magnetic field is applied to a conductor, creating an electric current within the conductor. This process is known as magnetic induction and is governed by the magnetic induction formula. In this article, we will explore the magnetic induction formula in detail, including its definition, application, and mathematical representation.
Magnetic induction refers to the phenomenon of magnetizing materials when exposed to an external magnetic field, resulting in the emergence of certain magnetic properties in the material. The relationship between electricity and magnetism has been understood for nearly 200 years, with scientists discovering that the movement of electric charges (i.e., electric current) generates magnetic fields. Conversely, the movement of magnets can also produce electric currents. Electromagnetic induction, also known as magnetic induction, is the process by which an electrical conductor placed within a varying magnetic field produces an electromotive force (EMF) or voltage.
The magnetic induction formula is given as: ϵ = dϕb/dt
Faraday's experiments led to the discovery that a coil experiences an induced emf when there is a time-varying magnetic flux passing through it. This principle is known as Faraday's Law, which states that the "rate of change of magnetic flux in a circuit produces an emf in that circuit." The magnitude of the emf induced in a circuit is equivalent to the rate of change of the magnetic flux passing through it.
The equation for electromotive force (EMF) in a closed circuit can be derived from Faraday's law and is represented as follows:
ϵ = - (dϕb / dt)
In this context, ϕb represents the magnetic flux, ϵ represents the electromotive force (EMF), and t is the time. The negative sign in the equation indicates that a current I and magnetic field B, opposite in direction to the change in flux, are generated. This concept is known as Lenz's law.
In the case of a closely wound coil comprising N turns, the change in flux associated with each turn is identical, resulting in the total induced EMF or magnetic induction being represented as:
ϵ = N(dϕb / dt)
For a moving rod, with N = 1 and the flux given by Φ = BAcosθ, where θ = 0º and cosθ = 1, the area swept out by the rod is ΔA = lΔx.
ϵ = BΔA/Δt
Here, velocity is perpendicular to the magnetic field.
If the velocity is at an angle θ with B, its component perpendicular to B is v sinθ.
ε = Blv sinθ
Here,
l = length of the conductor,
v = velocity of the conductor
θ = the angle between the magnetic field and the direction of motion.
Therefore, the induced current formula denotes a close relation between electric field and magnetic field that is dependent on a specific time variation.
Electricity and magnetism are concepts that are closely related. Almost 200 years ago, scientists discovered that moving electric charges (electric current) generated magnetic fields.
The magnetic flux, denoted by ΦB, is a measurement of the number of magnetic field lines that intersect a surface with cross-sectional area A. The amount of magnetic flux through a small surface is calculated by multiplying the magnetic flux density that is perpendicular to the surface by the area of the surface. Similar to electric flux, magnetic flux is defined as ΦB = B.AcosÏ•,
Here Ï• is the angle between B and A. Magnetic Flux is a scalar quantity. Its SI unit is Weber.
1 Weber = 1 Tesla.meter2.
According to Lenz's law, the direction of the induced emf is such that it generates a current that opposes the change in magnetic flux that caused it.
As shown in the figure below, the north pole of a bar magnet is being moved towards a closed coil, resulting in an increase in the magnetic flux through the coil. Consequently, the induced current in the coil flows in a direction that opposes the increase in flux, which is in a counterclockwise direction.
If we consider a straight conductor moving in a magnetic field, we can see in the figure that the rod PQ moves towards the left at a constant velocity v. As long as there is no energy loss, the circuit PQRS forms a closed loop that encloses an area changing as PQ moves. The magnetic flux can be expressed as ΦB = Blx, and since x changes with time, a rate of change of flux induces an emf given by:
ε = -dΦB/dt = -d/dt(Blx) = -Bld/dt = Blv.
The motional emf expression can also be explained by the Lorentz force acting on the free charge carriers of conductor PQ. As the rod moves with speed v in the magnetic field B, the charge also moves with the same speed v. The Lorentz force acting on this charge is qvB in magnitude and it is directed towards Q. The work done in moving the charge from P to Q is given by the product of the force and the distance moved, which is qvBL.
Since emf is defined as the work done per unit charge, it can be expressed as:
ϵ = W/q = BLv
The table below shows the magnetic induction for different values of current and distance:
|
Current (I) |
Distance (r) |
Magnetic Induction (B) |
|
1 A |
1 m |
2 x 10^-7 T |
|
2 A |
1 m |
4 x 10^-7 T |
|
1 A |
2 m |
1 x 10^-7 T |
|
2 A |
2 m |
2 x 10^-7 T |
As can be seen from the table, the magnetic induction increases with the current and decreases with the distance from the wire.
The magnetic induction formula is used in various applications, including:
Sol:
ϵ2/ϵ1 = N2/N1 (From Transformer Equation)
Where ϵ1 is the induced EMF in the primary coil, ϵ2 is the induced EMF in the secondary coil, N1 is the number of turns in the primary coil, and N2 is the number of turns in the secondary coil.
Also, ϵ1 = N1(dϕb/dt) (From Magnetic Induction Formula)
Given:
N1 = 500
N2 = 1000
dϕb/dt = 1000 T/s
Substituting the values, we get:
ϵ1 = 500 x (1000 T/s)
ϵ1 = 500000 V
Using the transformer equation, we get:
ϵ2/500000 = 1000/500
ϵ2 = (1000/500) x 500000
ϵ2 = 1000 V
Therefore, the induced EMF in the secondary coil is 1000 volts.
Sol:
l = 0.5cm
B = 0.6T
θ = 90°
v = 2 m/s.
E = Blv
= 0.5 × 0.6 × 2
= 0.6 V
Hence induced emf is 0.6Volt.
Sol:
B = 2 × 10-2 T
θ = 90°
A = 100cm2 = 0.01m2
n = 50 turns.
E = 0.1 V
Or, θ = B × A
= 2 × 10-2 × 0.01
= 0.0002
E = N. θt
Or, 0.1 = 50 × 0.002 t
So, t = 0.1 sec.
Sol:
i) Area (A) = 50 cm2 = 5 × 10-3 m2
B = 10-3T
Now,
(i) Flux(θ) = B × A
= 5 × 10-2 Tm2
ii) E = dθ/dt
= 5 × 10-6 − 0
= 1.67 × 10-6V
Sol:
ϵ = N (dϕb/dt) (From Magnetic Induction Formula)
Where N is the number of turns in the coil.
Given:
N = 200
dϕb/dt = 2 T/s
Ï•b = 5 x 10^-3 Wb
Substituting the values, we get:
ϵ = 200 x (2 T/s) x (5 x 10^-3 Wb)
ϵ = 2 V
Therefore, the induced EMF in the coil is 2 volts.
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By - Nikita Parmar 2023-08-25 09:48:44 , 11 min readThe Magnetic Induction Formula expresses the relationship between the induced electromotive force (EMF) and the rate of change of magnetic flux. It is given by ϵ = dϕb/dt, where ϵ is the induced EMF, ϕb is the magnetic flux, and dt is the change in time.
The unit of measurement for magnetic induction is the volt (V).
Lenz’s Law states that the polarity of the induced EMF is such that it tends to produce a current which opposes the change in magnetic flux that produces it.
The direction of the induced EMF is such that it opposes the change in magnetic flux that produces it, in accordance with Lenz’s Law.
The induced EMF is directly proportional to the number of turns in a coil, as given by the equation ϵ = N(dϕb/dt), where N is the number of turns.
The Motional EMF Formula expresses the induced EMF in a metal rod moving perpendicular to a uniform magnetic field. It is given by ϵ = Blv, where B is the magnetic field, l is the length of the rod, and v is the velocity of the rod.
The negative sign in the Magnetic Induction Formula indicates that the induced EMF opposes the change in magnetic flux that produces it, in accordance with Lenz’s Law.
Eddy currents are current loops that are set up in nearby metal (any conductor) bodies due to changing magnetic fields. They dissipate electrical energy as heat.
Magnetic induction is used in various electrical devices such as transformers and electric generators to generate electrical energy.
Magnetic flux is the measure of the amount of magnetic field passing through a surface, while magnetic field is the force exerted by a magnet on a magnetic object or charged particle. The Magnetic Induction Formula relates the two by expressing the induced EMF in a conductor placed inside a varying magnetic field.
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