Electromagnetic Induction notes

Magnetic Flux and Electromagnetic Induction

Magnetic flux is the total magnetic field passing normally through a surface. It is represented by Φ and is measured in weber (Wb). For a uniform magnetic field, magnetic flux is given by Φ = BA cos θ, where B is magnetic field strength, A is area of the coil, and θ is the angle between the magnetic field and the normal to the plane of the coil.

Electromagnetic induction is the phenomenon of production of induced electromotive force in a coil due to a change in magnetic flux linked with the coil. The change may be produced by moving a magnet, moving a coil, changing the area of the coil, changing its orientation, or changing the magnetic field strength.

• Electromagnetic induction means production of induced emf due to changing magnetic flux.
• Magnetic flux: Φ = BA cos θ.
• The SI unit of magnetic flux is weber (Wb).
• An induced emf exists only while the magnetic flux is changing.
• A stationary coil in a constant magnetic field has no induced emf.
• A coil may be moved in a magnetic field to produce electromagnetic induction.
• The time rate of change of magnetic flux has the same dimensions as potential difference, because dΦ/dt has the dimensions of volt.

Faraday's Laws of Electromagnetic Induction

Faraday's first law states that whenever the magnetic flux linked with a circuit changes, an emf is induced in the circuit. If the circuit is closed, an induced current also flows. If the circuit is open, emf is present but current does not flow.

Faraday's second law states that the magnitude of induced emf is equal to the rate of change of magnetic flux linkage. For a coil of N turns, emf is given by ε = -N ΔΦ/Δt. The negative sign represents Lenz's law. The magnitude is ε = N ΔΦ/Δt.

• The induced emf is proportional to the number of turns N.
• The induced emf increases when the magnetic flux changes more rapidly.
• The induced emf is not determined by the resistance of the coil.
• Increasing coil resistance does not change the induced emf.
• Current depends on resistance and is given by I = ε/R for a simple circuit.
• For one turn, induced emf magnitude is ε = ΔΦ/Δt.
• The term emf means electromotive force, although emf is not a mechanical force.
• Example: If L = 0.02 H and dI/dt = 150 A/s, induced emf ε = L(dI/dt) = 0.02 × 150 = 3 V.

Lenz's Law and Conservation of Energy

Lenz's law gives the direction of induced current. It states that the induced current always flows in such a direction that its magnetic field opposes the change in magnetic flux that produces it. This opposition does not mean that it opposes the original magnetic field in every situation. It opposes the change in flux.

The negative sign in Faraday's law represents Lenz's law. Lenz's law is based on the law of conservation of energy. External work must be done to move a magnet or coil against the opposing magnetic effect. This mechanical work is converted into electrical energy.

• Lenz's law is based on conservation of energy.
• If flux through a coil increases, the induced field opposes the increase.
• If flux through a coil decreases, the induced field supports the original field to oppose the decrease.
• A magnet approaching a coil produces a current whose magnetic pole repels the approaching pole.
• A magnet moving away produces a current whose magnetic pole attracts the receding pole.
• For the metallic ring falling toward the bar magnet, the induced current is clockwise in the direction shown in the given arrangement.
• The direction of induced current can be found using Lenz's law and the right-hand rule.

Generators, Motors, and Back Emf

An electric generator, or dynamo, converts mechanical energy into electrical energy by electromagnetic induction. A coil rotates in a magnetic field, so the magnetic flux linked with it changes continuously. An alternating current generator produces an alternating emf.

The induced emf in a rotating coil is maximum when the plane of the coil is parallel to the magnetic field. In this position, the magnetic flux itself is zero, but the rate of change of flux is maximum. The emf is zero when the plane of the coil is perpendicular to the magnetic field, because the rate of change of flux is then momentarily zero.

An electric motor converts electrical energy into mechanical energy. A motor can work as a generator or dynamo if it is driven mechanically. When a motor rotates, it also produces a back emf. This emf opposes the supply voltage and limits the current drawn by the motor.

• A generator works on electromagnetic induction.
• An alternating current generator changes mechanical energy into electrical energy.
• The induced emf is maximum when the plane of the rotating coil is parallel to the magnetic field.
• The induced emf is zero when the plane of the coil is perpendicular to the magnetic field at the relevant instant.
• An electric motor can be used as a dynamo when it is driven mechanically.
• Back emf opposes the supply voltage in a motor.
• Back emf limits the current drawn by a running motor.
• At starting, the motor has little or no back emf, so the starting current can be large.

Self Induction and Inductors

Self induction is the production of an induced emf in a coil due to a change in current through the same coil. The changing current changes the magnetic flux produced by the coil. The induced emf opposes the change in current according to Lenz's law.

The constant of proportionality between flux linkage and current is called self inductance or inductance. It is represented by L and is measured in henry (H). The induced emf due to self induction is ε = -L dI/dt. A coil with large inductance strongly opposes changes in current.

• Self induction occurs in one coil because of a change in its own current.
• Self inductance is represented by L.
• The SI unit of inductance is henry (H).
• For a coil, L = NΦ/I when flux linkage is proportional to current.
• For a long air-cored or iron-cored solenoid, L = μN²A/l.
• Inductance depends on the number of turns, cross-sectional area, length, and magnetic permeability of the core.
• Self inductance does not depend upon the weight of the coil.
• An iron core increases inductance because iron has high magnetic permeability.
• The energy stored in an inductor is U = 1/2 LI².
• An inductor stores energy in its magnetic field.

Mutual Induction and Eddy Currents

Mutual induction is the production of an induced emf in one coil when the current in a nearby coil changes. The coil connected to the battery and rheostat is called the primary coil. The other coil, in which emf is induced, is called the secondary coil. The coils are electrically separate but are magnetically linked through their common changing magnetic field.

A rheostat changes the current in the primary coil. When the primary current changes, the magnetic flux through the secondary coil changes, producing an induced emf. The practical application of mutual induction is the transformer.

Eddy currents are circular currents induced in bulk conductors when the magnetic flux through them changes. They can cause unwanted heating and energy loss, but they are also used in electromagnetic braking and induction heating.

• Mutual induction involves two separate coils.
• The primary coil contains a battery and a rheostat when mutual induction is studied.
• The secondary coil receives induced emf due to changing flux from the primary coil.
• There is no electrical connection between the two transformer coils, but they are magnetically linked.
• A magnet dropped through a long vertical copper tube falls slowly because induced eddy currents produce a magnetic field opposing the magnet's motion.
• Eddy currents oppose the change that produces them, in accordance with Lenz's law.
• Laminating an iron core reduces unwanted eddy current losses in transformers.
• Eddy currents can be useful in magnetic braking.

Transformers

A transformer is a device that changes alternating voltage and current using mutual induction. It has a primary coil and a secondary coil wound on a soft iron core. An alternating current in the primary produces changing magnetic flux in the core. This changing flux induces an emf in the secondary coil.

For an ideal transformer, the voltage ratio is related to the turns ratio by Vs/Vp = Ns/Np. If Ns is greater than Np, the transformer is a step-up transformer and increases voltage. If Ns is less than Np, it is a step-down transformer and decreases voltage.

For an ideal transformer, input power equals output power, so VpIp = VsIs. Therefore, when voltage is increased, current is decreased in the same proportion. A practical transformer has losses, so output power is less than input power.

• A transformer works on mutual induction.
• A transformer changes voltage and current, not resistance or capacitance.
• A transformer requires alternating current or another changing current.
• A transformer cannot be used with a direct current supply because steady direct current produces no changing flux and no induced emf in the secondary.
• Step-up transformer: Ns > Np and Vs > Vp.
• Step-down transformer: Ns < Np and Vs < Vp.
• Ideal transformer relation: Vs/Vp = Ns/Np.
• Ideal power relation: VpIp = VsIs.
• In a practical transformer, output power is less than input power.
• Transformer losses include copper loss, eddy current loss, hysteresis loss, and flux leakage.
• A transformer does not provide a direct electrical connection between its two coils.

Inductive Reactance, Heating, and Energy Relations

An inductor opposes changes in alternating current because its current is continuously changing. In an AC circuit, the opposition offered by an inductor is called inductive reactance, represented by XL. It is given by XL = 2πfL, where f is frequency and L is inductance. Like resistance, inductive reactance is measured in ohms, but it is caused by induction and does not represent ordinary energy loss in an ideal inductor.

A resistor converts electrical energy into heat. The power loss in a resistance is P = I²R. An ideal inductor stores energy in its magnetic field and returns that energy to the circuit, so it has no average power loss. Real coils have resistance and therefore produce some heat.

• Inductive reactance is XL = 2πfL.
• The unit of inductive reactance is ohm (Ω).
• Inductive reactance increases when frequency increases.
• Inductive reactance increases when inductance increases.
• An inductor behaves like an opposition similar to resistance in an AC circuit, but it stores energy magnetically.
• Energy stored in an inductor is U = 1/2 LI².
• Power loss in a resistor is P = I²R.
• Resistance affects current in a circuit but does not affect the induced emf produced by a specified rate of change of flux.
• The induced emf in a coil may be called back emf when it opposes a change in current in that coil.