Electromagnetic Induction Lecture Notes PDF

Summary

These lecture notes cover electromagnetic induction, including Lenz's and Faraday's experiment, magnetic flux, and various scenarios involving different configurations of loops and coils.

Full Transcript

# Electromagnetic Induction ## Lenz's and Faraday's Experiment - **When the current in the coil is zero (`Iin = 0`)**: - The magnetic field is constant (`B = Cost`) - The magnetic flux is constant (`Φ = Cost`) - **When the current in the coil is changing (`Iin ≠ 0`)**: - The magneti...

# Electromagnetic Induction ## Lenz's and Faraday's Experiment - **When the current in the coil is zero (`Iin = 0`)**: - The magnetic field is constant (`B = Cost`) - The magnetic flux is constant (`Φ = Cost`) - **When the current in the coil is changing (`Iin ≠ 0`)**: - The magnetic field is changing (`B = change`) - The magnetic flux is changing (`Φ = change`) - **The area of the loop is also changing when the current in the coil is changing**: - The magnetic flux is changing (`Φ = change`) - The magnetic field is changing (`B = Cost`) ## Magnetic Flux Like electric flux, magnetic flux is proportional to the number of magnetic field lines passing through a surface. It is denoted by `ΦB`. Mathematically, `ΦB = B * A = BAcosθ` - **Gives the idea of magnetic energy!** - **Current Induced in a Coil due to Change in Magnetic Flux**: - Due to change in Area - Due to change in Magnetic Field (`B`) - Due to change in Angle - **SI unit of magnetic flux is weber (Wb)**. 1Wb = 1 tesla m² - **C.G.S unit of magnetic flux is maxwell (1 maxwell = 1 gauss cm²)** - 1 weber = 10⁸ maxwell - **Magnetic flux (like electric flux) is a scalar quantity**. - **Magnetic flux can be calculated by Integration.** `ΦB = ∫B. dA` **The dimensional formula of magnetic flux is [ML²T⁻²A⁻¹]** `Φ = (F/q) A = MLT⁻² * L² / AT = [ML²T⁻²A⁻¹]` `Φ = B * A = (Tesla * *m²) = SI unit` ↳ **No direction** ## Magnetic Flux Through Different Areas - **Magnetic flux through an area 𝛖 if the area 𝛖 is in x-y plane**: `Φ = B * A = B * (S * k) = -B₃S` - **Magnetic flux through an area 𝛖 if the area 𝛖 is in y-z plane**: `Φ = B * A = B * (S * i) = B₁S` - **Magnetic flux through an area 𝛖 if the area 𝛖 is in z-x plane**: `Φ = B * A = B * (S * j) = B₂S` ## Lenz's Law Lenz's law states that the direction of the induced current in a circuit is such that it always opposes the change in magnetic flux that produced it. - **Based on conservation of energy** - **Only gives the direction of induced current** - **The direction of the induced current is such that it opposes the cause due to which it is induced** ## Faraday's Law of Induction The magnitude of the induced EMF in a circuit is equal to the time rate of change of magnetic flux through the circuit. - Mathematically, instantaneous EMF (i.e., EMF induced at time t = ts) `e(t) = dΦ(t) / dt` ## Induced EMF and Current in Different Scenarios **Scenario 1**: - **A loop is placed perpendicular to the current carrying wire.** `Φ = 0` **Scenario 2**: - **A loop is placed in front of a current carrying wire.** - `Iinduced = 0` - `Φ = 0 always` **Scenario 3**: - **A current carrying circular loop is placed in the x-y plane.** `Φxy plan = 0` **Scenario 4**: - **A rectangular loop is placed parallel to the axis of a solenoid.** `Φ = 0` **Scenario 5**: - **Find Flux Through Ring**: `ΦRing (r<<<l) = (2√2μ₀I/πl) * πr² = 2√2μ₀Ir²/l` `ΦSmall loop (r<<<R) = BA = (μ₀I/2πR) * πr² = μ₀Ir²/2R` **Scenario 6**: - **Find flux through square loop**: `Bcl = (μ₀I/2πx) - variable` `dΦ = B dx l = (μ₀I/2πx) dx l` `Φ = (μ₀I/2π) ∫(1/x) dx = (μ₀I/2π) * ln(a+l/a)` `Φcomp Ring = BAR²/2` **Scenario 7**: - **A small rectangular loop is moving towards left with constant velocity through a uniform B field.** `Φ = Blx` `dΦ/dt = Bl(dx/dt) = BlV` `e.m.f = dΦ/dt = BlV = Coul` - **The variation of flux through the loop with respect to time**: - The flux increases linearly from initially zero when the loop enters the magnetic field until it reaches the full area when the loop enters fully. Then it remains constant. - **The variation of induced emf w.r.t. time**: - The induced emf remains constant when the loop is entering the magnetic field. Then it becomes zero when the loop enters fully. **Scenario 8**: - **A loop of irregular shape of conducting wire PQRS changes into a circular shape.** `I=Anti-c` `Φ(B○T)` - **The direction of induced current will be Anti-clockwise.** **Scenario 9**: - **A conducting wire XY is moved towards the right, a current flows in the anti-clockwise direction.** - **Direction of magnetic field at point O is Perpendicular outside the paper.** **Scenario 10**: - **An electron moves on a straight line path XY. The abcd is a coil adjacent to the path of the electron.** - **The current will reverse its direction as the electron goes past the coil.** **Scenario 11**: - **A bar magnet is dropped through a horizontal aluminium ring along the axis of the ring.** - **The direction of induced current for the observer shown is Anti-clockwise.** - **The direction of a magnetic force experienced by the bar magnet is upwards.** **Scenario 12**: - **A magnet N-S is suspended from a spring and when it oscillates, the magnet moves in and out of the coil C.** - **The galvanometer G shows deflection to the left and right but the amplitude steadily decreases.** **Scenario 13**: - **There are two loops A and B placed coaxially along the vertical line. Loop A is allowed to fall freely towards loop B.** - **The direction of flow of induced current in loop B as seen from the bottom of loop B is clockwise.** **Scenario 14**: - **A metallic ring with a cut is held horizontally and a magnet is allowed to fall vertically through the ring.** - **The acceleration of this magnet is Equal to g.** **Scenario 15**: - **A short bar magnet passes at a steady speed right through a long solenoid. A galvanometer is connected across the solenoid.** - **The graph represents the variation of the galvanometer deflection 0 with time t is the one where the deflection reaches the maximum first then becomes zero, then again reaches the same maximum in the opposite direction and finally becomes zero again.** **Scenario 16**: - **A coil having 500 square loops each of side 10 cm is placed with its plane perpendicular to a magnetic field which increases at a rate of 1.0 tesla/s.** - **The induced e.m.f. (in volts) is 5.0.** **Scenario 17**: - **Radius of a circular loop placed in a perpendicular uniform magnetic field is increasing at a constant rate of r₀ ms¯¹.** - **The emf induced in the loop at that instant is -2Bπr₀r.** **Scenario 18**: - **A conducting circular loop is placed in a uniform magnetic field, B = 0.025 T with its plane perpendicular to the loop.** - **The radius of the loop is made to shrink at a constant rate of 1 mm s¯¹.** - **The induced emf when the radius is 2 cm is πμV.** **Scenario 19**: - **In a coil of resistance of 10Ω, the induced current developed by changing magnetic flux through it, is shown in a figure as a function of time.** - **The magnitude of change in flux through the coil in Weber is 2.** **Scenario 20**: - **The magnetic flux through a circuit of resistance R changes by an amount ΔΦ in a time Δt.** - **The total quantity of electric charge Q that passes any point in the circuit during the time Δt is represented by Q= ΔΦ/R**

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