Electromagnetic Induction Lecture Notes PDF
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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.
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# 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**