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Electromagnetic induction: When an electric charge moves in a magnetic field, then a force acts on it. In a reverse sense, a movement or change in magnetic field relative to stationary charge gives raise to an electric current.

 

Induced emf (ε)

  • Definition: Potential difference generated by electromagnetic induction.

For a rod of length L moved with velocity v in a region of magnetic field B:

  • If ​the rod moves from left to right, and thus, its electrons move perpendicular to the magnetic field, they experience a downward force along the rod and an electric field is established.

  • Flow of electrons quickly stops due to electrostatic repulsion at the bottom, and thus, the current exists for a short period of time.

  • Without movement, emf is not induced.

  • Formula if the rod is moved connected to wires (the work done to separate electrons leads to an induced emf): ε = BvL.

 

Magnetic flux (Ф)

  • Definition: "Product of the magnitude of the normal component of magnetic field strength and area through which it passes."

  • Intuitive picture: Number of magnetic field lines crossing a certain area.

  • Formula: Ф = BAcosθ, where A is the area and θ is the angle between the magnetic field strength direction and the direction normal to the loop area.

  • Units: weber (Wb)

  • Definition for a rod: "Product of magnitude and the rate at which the area swept out by the rod is changing" = ∆Ф/t.

  • Magnetic flux linkage: Magnetic flux multiplied by the N turns in a loop. Ф = NBAcosθ.

  • Magnetic flux density: numerically equivalent to magnetic field strength.

    • Induced emf = magnetic flux density x rate of change of area =​ BA/t.

 

Faraday's Law

  • Definition: "Induced emf is equal to the negative rate of change of magnetic flux linkage."

    • Negative sign exists due to Lenz's law (see below).​

  • Formula: ε = -NФ/t.

  • Rod (perpendicular to field): in time ∆t, a rod of length L will move a distance s = v∆t, cutting magnetic field lines as it moves in the magnetic field. A = Ls

    • Formula: ∆Ф = ∆BAcos0º = ∆BA = ∆BLs = BLv∆t, and hence, ε = BvL.

 

Lenz's Law

  • Definition: "The induced emf will be in such a direction to oppose the change in the magnetic flux that created the current​". It is equivalent to energy conservation.

  • Work done by magnetic forces that arises due to current is dissipated as thermal energy.

Examples:

  • Rod: Force in the rod must oppose the motion. Hence, if it moves towards the right, a leftwards force will appear indicating a counter-clockwise induced current. ​

    • Use left-hand rule twice: Firstly to find the direction of the current in the loop. Secondly, to find the force induced on the rod due to the current.

  • Loop wire and a wire with increasing current: Magnetic flux is increasing into the page. Hence, to oppose the increase in magnetic flux (inside the loop), a magnetic field out of the page must exist, and thus, a counter-clockwise current is induced.​

(Tsokos, 2014)

  • Bar magnet through a loop of wire:

    • When approaching the loop, magnetic flux is increasing, and thus, magnetic field must oppose the increase, with a counter-clockwise current.

    • When leaving the loop, the magnetic flux is decreasing, and the current is now clockwise. ​

The opposite magnet (south pole first) would have the exact opposite effect.

The alternating current (ac) generator

  • Rotating coil in a region of magnetic field.

  • A magnetic field that cuts the rotating coil.

  • Relative movement between the coil and the magnetic field, causing emf to be induced and current to flow.

    • Rotation of coil: caused by a turbine in a power plant​.

  • Two slip rings attached to the ends of the coil, rotating along with it and touching carbon brushes that transfer current to the outside world.

  • Current: When the left-hand wire is moving upwards and the right-hand wire is moving downwards, current is counter-clockwise. Half a period later, the current will be opposite.

  • Induced emf, by Faraday's law, is the minus rate of change of the flux linkage.

    • If the angle speed (w) increases, frequency and emf amplitude increase.

    • Increasing N, B or A causes the emf to increase, without changing the frequency

  • Power in ac circuits always positive, with a period of half the frequency.

    • Average power dissipated: half the peak's value.​

    • Same phase as current and emf.

    • π/2 out of phase with the graph of change of magnetic flux linkage (it is its derivative)

Root mean square (rms)

In order to fund the average value of the current and of the emf (given that they are both negative and positive), we the root of the peak value divided by 2.

  • Current rms = Irms = sqrt(Io²/2)

  • Voltage rms = Vrms = sqrt(Vo²/2)

  • Mean power = (Irms)(Vrms)/2 Irms²R/2 = Vrms²/(2R)

 

The transformer

Changes the potential difference from one alternating current into another potential difference.

  • Process: 

  1. Alternating current produces magnetic field in primary coil.

  2. Flux in the core is created.

  3. Changing flux is linked to the secondary coil.

  4. If coil is part of a circuit, current flows.

  • Formulas: εps = Np/ Ns = Vp/Vs = Is/Ip. (primary = p; secondary = s).

    • N = number of coils.

    • Frequency remains unchanged.

  • Step-up transformer: Output voltage > Input voltage: Ns > Np.

  • Step-down transformer: Output voltage < Input voltage: Ns < Np.

  • Graph of secondary coil = gradient of graph of primary coil.

Real transformers:

  • Core material: soft magnetic material (avoids magnetic hysteresis). 

    • Can be rapidly magnetized and demagnetized.​

  • Core design/shape: ensures flux does not leak out of the core (less power loss)​.

  • Laminators: prevent the formation of currents inside the core itself, known as eddy currents, which lead to heating and power loss.

Transformers and power transmission

  • Power loss in cables proportional to (current)².

  • Reducing power loss: If pd is increased by transformer then I is decreased, so power falls, as R is constant.

    • Other benefits: Smaller I - Smaller temperature - Smaller R - Lower genetic damage.​

Rectification of alternating current

Process of converting an alternating current supply into direct current.

  • Rectifier: a diode, which only allows current to pass in one direction.

    • When current passes through it, the diode is said to be forward biased.

    • When no current passes through it, it is reverse biased.

Half-wave rectification: half of the power is lost.​

  • Current and voltage is not constant: It is zero during half a period.​

Full-wave rectification: usage of the whole power.

  • Two different cycles, as shown below: Forward Bias and Reverse Bias.

Rectification with capacitors: using a circuit with a capacitor (in parallel with resistor), which charges when alternate current is forward and discharges with reverse bias.

  • Useful to overcome the problem of zero current, by creating small ripples.

Topic 11: Electromagnetic Induction

11.1 Electromagnetic Induction

11.2 Transmission of Power

11.3 Capacitance

Capacitors

Capacitor: Two conductors separated from each other by an insulating (dielectric) material (or vacuum).

  • Storage: Stores electric charge and electric energy.

  • Design: Formed by two parallel plates with area and distance in between.

  • Capacitance: charge (q) per unit voltage (V) that can be store in a capacitor

    • Charge distribution: +q on one plate and -q on the other plate

    • C = Q/V = εA/d

    • Units: Farad (F)

In a Closed Circuit

  • Capacitor: 

    • Electrons move from the plate connected to the positive terminal and transfer to the plate connected to the negative terminal.

    • Potential difference across capacitor is greater or equal to emf across it.

  • Combining capacitors:​ Opposite as with resistors!

    • In parallel: ∑C = C1 + C2​ + C3 +... (same pd across them)

    • In series: 1/∑C = 1/C1 + 1/C2​ + 1/C3 +... (same charge across them)

  • Energy stored:​ total work done to charge the capacitor

    • E = 1/2 CV² = 1/2 QV = 1/2 Q²/C

The effect of dielectric

  • Dielectric material: ε > εo (vacuum), and thus, C > Co

  • Charge polarization:​ In the dielectric, there is separation of charges, known as charge polarization.

    • Small electric field is created, reducing the net electric​ compared to εo

    • pd across capacitor is also reduced, since some electric energy is used to align molecules, raising the potential of the negative plate and lowers the potential of the positive plate

Charging and discharging

  • Charging: Accumulating charge on the negative plate

    • Current starts out large, as if the capacitor was not there, i.e Io = ε/R, but decreases and reaches zero, since the electrons on the negative plate push back new electrons. 

    • When fully charged, no current passes through the capacitor.

  • Discharging: Capacitor becomes a power source, which is discharge by resistors

    • Formulas: q = qo e^-t/RC​; V = Vo e^-t/RC​; I = Io e^-t/RC​ and Io = qo/RC

    • Time constant (τ) = RC, is the time scale for discharge (measured in seconds).

      • Time took for q to decrease to 37% of its original value as it discharges

      • Similar to half-life in radioactive decay.