Electromagnetic Induction

For a rectangular loop with length l being pulled from magnetic field B with velocity v (resistance R):

1. Induced EMF: $\mathcal{E} = vBl$ • From Faraday's Law: $\mathcal{E} = -\frac{d\Phi}{dt} = -B\frac{dx}{dt}l = vBl$

2. Magnetic force: $F_1 = \frac{B^2l^2v}{R}$ (opposite to velocity) • Current: $i = \frac{\mathcal{E}}{R} = \frac{vBl}{R}$ • Force: $F_1 = ilB = \frac{vBl}{R} \cdot l \cdot B = \frac{B^2l^2v}{R}$

3. External force needed: $F = F_1 = \frac{B^2l^2v}{R}$ • To maintain constant velocity, must exactly counter magnetic force

4. Power calculations: • Power delivered: $P = Fv = \frac{v^2B^2l^2}{R}$ • Thermal power: $P = i^2R = (\frac{vBl}{R})^2R = \frac{v^2B^2l^2}{R}$

These are equal by conservation of energy: all mechanical work done against magnetic force is converted to thermal energy in the resistor. This demonstrates perfect energy conversion between mechanical and electrical systems with no loss.

Lenz's Law states: The direction of induced current is such that it opposes the change that has induced it.

Application rules:

• If flux is increasing: induced current creates a magnetic field opposing the original field
• If flux is decreasing: induced current creates a magnetic field reinforcing the original field

For example:

1. When a north pole approaches a loop, the induced current creates a magnetic field pointing toward the magnet (effectively creating a "north pole" facing the approaching magnet)
2. When a conducting loop is being pulled out of a magnetic field, the induced current creates a magnetic field that tries to maintain the original flux

This principle is a manifestation of energy conservation - the induced current always flows in a direction that opposes the motion or change causing it.

Motional EMF is the electromotive force induced in a conductor moving through a magnetic field.

Key characteristics:

• Generated when a conductor cuts through magnetic field lines
• Caused by magnetic force separating charges within the conductor
• Formula: E = vBl where:
  • v = velocity of conductor (perpendicular to field)
  • B = magnetic field strength
  • l = length of conductor in the field

Mechanism:

1. Free electrons in the moving conductor experience a magnetic force F = qv × B
2. This force drives electrons toward one end, creating charge separation
3. Electric field builds up until magnetic and electric forces balance (qE = qvB)
4. Resulting potential difference across the conductor is E = vBl

This principle is the foundation of electrical generators where mechanical motion is converted to electrical energy.

• Lenz's law application: The induced current creates a magnetic field that opposes the change causing it
• When pulling loop out: The decreasing magnetic flux through the loop induces current to strengthen the original flux
• Current direction: Clockwise when viewed from above (to create magnetic field into the page)
• Opposing force magnitude: F = B²l²v/R where:
  • B = magnetic field strength
  • l = length of the wire segment in the field
  • v = velocity of the loop
  • R = total resistance of the loop
• Physical interpretation: The higher the velocity, field strength, or length of wire in the field, the greater the opposing force
• Energy perspective: The mechanical work done against this opposing force is converted to thermal energy in the loop

Slotting a metal plate:

• Interrupts the potential circular paths for eddy currents
• Forces currents to flow through longer, more resistive paths
• Reduces the overall magnitude of eddy currents
• Decreases the heating effect and energy losses

This technique is used in electrical machinery because:

• It significantly reduces electromagnetic damping
• Minimizes energy losses due to eddy current heating (I²R losses)
• Improves efficiency of transformers, motors, and generators
• Allows for faster changes in magnetic fields without excessive heating

Applications:

1. Transformer cores are made of thin laminated sheets insulated from each other
2. Armature cores in motors have slotted designs
3. Induction furnace components use segmented plates
4. Magnetic braking systems utilize controlled eddy current formation

Note: While completely eliminating eddy currents is impossible in conductive materials experiencing changing magnetic fields, slotting reduces their effects to manageable levels.

Energy balance in an LR circuit:

1. The work done by the battery during current growth equals: $W_{battery} = \int_0^t \mathcal{E}i\,dt$

2. This energy is distributed as:

   • Energy dissipated in the resistor: $\int_0^t i^2R\,dt$
   • Energy stored in the inductor's magnetic field: $\frac{1}{2}Li^2$

3. At any time $t$: $\int_0^t \mathcal{E}i\,dt = \int_0^t i^2R\,dt + \frac{1}{2}Li^2$

Energy storage:

• The inductor stores energy in its magnetic field
• Energy density in the magnetic field: $u = \frac{B^2}{2\mu_0}$ (joules/m³)
• When current reaches steady state, all subsequent energy from the battery is dissipated as heat in the resistor
• The maximum energy stored in the inductor at steady state is $\frac{1}{2}L(\frac{\mathcal{E}}{R})^2$

The current decay in an LR circuit follows the exponential function:

$i = i_0 e^{-t/\tau}$

Where:

• $i$ is the current at time $t$
• $i_0$ is the initial current
• $\tau = L/R$ is the time constant

After one time constant ($t = \tau$), the current decreases to: $i = i_0 e^{-1} = 0.37i_0$

This means 37% of the initial current remains, or equivalently, 63% of the current has decayed after one time constant.

When current is established in an LR circuit:

• The total energy delivered by the battery can be calculated as: ∫εi dt

• This energy is distributed as:

  • Energy stored in the inductor: ½Li²
  • Energy dissipated as heat in the resistor: ∫i²R dt

• During current growth, the battery must deliver enough energy to:

  1. Overcome the back emf (-L·di/dt) produced by the inductor
  2. Supply the resistive losses in the circuit

• When steady state is reached, all additional energy from the battery becomes heat in the resistor, as the stored magnetic energy remains constant

Growth vs. Decay in primary circuit:

Growth (circuit closed):

• Current increases slowly due to self-inductance
• Low rate of change (di/dt)
• Produces small induced EMF in secondary

Decay (circuit broken):

• Current decreases rapidly, approaching instantaneous
• Very high rate of change (di/dt)
• Produces large induced EMF in secondary

This asymmetry is crucial because the induction coil's purpose is generating high voltages in the secondary, which depends entirely on maximizing di/dt during current decay. Without this asymmetry, the device couldn't produce the high voltage spikes needed for practical applications.

Step 1: Calculate area of square loop

• Side length = 40 cm ÷ 4 = 10 cm
• Area = 10² cm² = 0.01 m²

Step 2: Calculate induced EMF

• $\mathcal{E} = A\frac{dB}{dt} = (0.01 \text{ m}^2)(0.02 \text{ T/s}) = 2 \times 10^{-4} \text{ V}$

Step 3: Calculate resistance

• $R = \rho\frac{l}{A_\text{wire}} = (1.7 \times 10^{-8} \Omega\text{m})\frac{0.4 \text{ m}}{\pi \times 10^{-6} \text{ m}^2} = 2.16 \times 10^{-3} \Omega$

Step 4: Calculate current

• $I = \frac{\mathcal{E}}{R} = \frac{2 \times 10^{-4}}{2.16 \times 10^{-3}} = 9.3 \times 10^{-2} \text{ A}$

Parameters affecting current: loop area, rate of field change, wire length, wire cross-sectional area, and material resistivity.