Electromagnetic Induction

Two mechanisms of electromagnetic induction:

1. Motional EMF: • Occurs when a conductor moves through a static magnetic field • EMF caused by magnetic force on free charges in the conductor: E = vBl • Magnetic force (qv×B) separates charges, creating potential difference • Dominates when: conductor moves but B-field is static • Example: Generator with moving wire in fixed magnetic field

2. Induced Electric Field: • Occurs when magnetic field changes with time at a fixed location • Non-conservative electric field created by changing B-field • Closed field lines with no starting/ending points • Dominates when: B-field changes but conductor is stationary • Example: Transformer with stationary coils

The distinction is fundamental: motional EMF requires moving conductor and works via Lorentz force, while induced electric field requires changing field and exists independently of any conductor.

The capacitor serves three critical functions:

1. Prevents spark formation at the contact points by providing an alternative path for current
2. Accelerates current decay rate (increases di/dt) when circuit breaks, enhancing induced secondary EMF
3. Creates current reversal by discharging through primary in opposite direction, changing current from i to approximately -i

This nearly doubles the magnetic flux change, significantly increasing the secondary EMF while simultaneously protecting the mechanical contacts from damage.

Direction of induced current when wire current increases:

Mathematical approach:

1. Choose a positive sense (direction) for current in the loop
2. Find positive normal direction using right-hand thumb rule (fingers curl along loop, thumb points along normal)
3. Calculate magnetic flux: Φ = ∫B⃗·dS⃗ (flux is positive if B and normal are aligned)
4. Determine if flux is increasing or decreasing:
   • When wire current increases, B-field and flux increase
   • So dΦ/dt is positive
5. Apply Faraday's Law: E = -dΦ/dt
   • If dΦ/dt > 0, then E < 0
   • Current flows opposite to chosen positive direction

Lenz's Law approach:

1. Identify the external change (wire current increasing)
2. Determine resulting magnetic field change (B-field strengthening)
3. The induced current must oppose this change:
   • Induced current creates its own B-field
   • This field must weaken the increasing field from the wire
   • Induced current flows to create a field opposing the original field

Example: For a loop to the right of a vertical wire with upward current, the induced current will flow clockwise when the wire current increases.

• When a bar magnet is moved toward a conducting loop, the galvanometer shows a deflection, indicating an electric current is flowing in the loop
• This occurs because the magnetic flux through the loop increases as the magnet approaches
• According to Faraday's Law: E = -dΦ/dt (where Φ is magnetic flux)
• The negative sign indicates the induced current creates a magnetic field opposing the change that produced it
• If the magnet stops moving, the flux stops changing, and the current disappears
• Example application: This principle is used in electrical generators where mechanical motion is converted to electrical current

To determine the direction of induced current when a rectangular loop's area decreases in a magnetic field:

1. Identify the change: Decreasing area means decreasing magnetic flux through the loop
2. Apply Lenz's Law: The induced current must create a magnetic field that opposes this decrease
3. Since the original flux is decreasing, the induced current must produce a magnetic field in the same direction as the original field (into the plane)
4. Using the right-hand rule: When your thumb points in the direction of the induced magnetic field (into the plane), your fingers curl in the direction of the current

Therefore, the induced current will flow clockwise around the rectangular loop PQRS when viewed from the direction the magnetic field is pointing.

This creates a magnetic attraction that opposes the physical reduction of the loop's area.

Solution process:

1. Identify the components for the motional EMF formula:

   • Length (l) = 25 cm = 0.25 m
   • Velocity (v) = 2 m/s
   • Magnetic field (B) = 0.5 T
   • Total resistance (R) = 2.5 Ω

2. Calculate the induced EMF: EMF = vBl = (2 m/s)(0.5 T)(0.25 m) = 0.25 V

3. Calculate the induced current using Ohm's law: I = EMF/R = 0.25 V/2.5 Ω = 0.1 A

The current flows in a direction that creates a magnetic force opposing the motion (according to Lenz's law). If the rod is moving to the right and the magnetic field points into the page, the current flows clockwise in the circuit.

For a shrinking circular loop in a uniform magnetic field B perpendicular to the loop:

1. Magnetic flux: $\Phi = \pi r^2 B$
2. Rate of change of flux: $\frac{d\Phi}{dt} = 2\pi r B \frac{dr}{dt}$
3. Induced EMF: $\mathcal{E} = -\frac{d\Phi}{dt} = -2\pi r B \frac{dr}{dt}$

The negative sign indicates the EMF opposes the change causing it (Lenz's law). The EMF depends on:

• The instantaneous radius (r)
• Field strength (B)
• Rate of radius change (dr/dt)

The time constant τ in an LR circuit:

• Is defined as τ = L/R (inductance divided by resistance)
• Physically represents how quickly energy stored in the magnetic field dissipates
• Determines the rate of exponential current decay: i = i₀e^(-t/τ)
• Larger τ means slower decay rate (longer transient response)

Effects of component changes:

• Increasing inductance (L): Larger τ, slower decay rate
• Increasing resistance (R): Smaller τ, faster decay rate

This behavior occurs because:

1. Higher inductance means more stored magnetic energy to dissipate
2. Higher resistance means faster energy dissipation through heat

In practical applications, engineers select L and R values to achieve desired transient response characteristics.

For magnetic field $B = (0.20 \text{ T})\sin[(50\pi \text{ s}^{-1})t]$ and loop area A:

1. Magnetic flux: $\Phi(t) = BA\sin[(50\pi \text{ s}^{-1})t]$

2. Induced EMF: $\mathcal{E}(t) = -\frac{d\Phi}{dt} = -BA\frac{d}{dt}\sin[(50\pi \text{ s}^{-1})t]$ $\mathcal{E}(t) = -BA(50\pi \text{ s}^{-1})\cos[(50\pi \text{ s}^{-1})t]$ $\mathcal{E}(t) = -10\pi BA\cos[(50\pi \text{ s}^{-1})t]$

3. Induced current (for resistance R): $i(t) = \frac{\mathcal{E}(t)}{R} = -\frac{10\pi BA}{R}\cos[(50\pi \text{ s}^{-1})t]$

The EMF and current oscillate at the same frequency as the magnetic field but are 90° out of phase.

An induction coil transforms low-voltage DC to high-voltage pulses through:

1. Electromagnetic induction: Changing current in primary creates changing magnetic flux

2. Rapid current interruption: Mechanical "make and break" system creates asymmetric current changes (slow rise, rapid fall)

3. Rate enhancement: Capacitor accelerates current decay rate and creates current reversal, maximizing di/dt

4. Turn ratio advantage: Secondary coil typically has many more turns than primary (NS >> NP)

5. Flux concentration: Iron core maximizes magnetic flux linkage between coils

The combination of these factors allows a 12V DC source to produce pulses exceeding 50,000V, with voltage gain primarily determined by the turn ratio and the enhanced rate of change in the primary current.