Solve Electromagnetic Induction

手順

ステップ1: Identify Source of Changing Flux

Classify the physical mechanism that produces a time-varying magnetic flux:

1. Changing B-field: The magnetic field itself varies in time (e.g., AC electromagnet, approaching magnet, current ramp in a nearby coil). The loop is stationary.
2. Changing area: The loop area changes (e.g., expanding or contracting loop, rotating coil in a static field). The B-field may be static.
3. Moving conductor (motional EMF): A straight conductor moves through a static B-field. The flux change arises from the conductor sweeping out area.
4. Combined: Both the field and geometry change simultaneously (e.g., a coil rotating in a time-varying field). Separate the contributions for clarity.

For each mechanism, identify the relevant surface S bounded by the circuit loop C:

## Flux Change Classification
- **Mechanism**: [changing B / changing area / motional / combined]
- **Surface S**: [description of the surface bounded by the loop]
- **Time dependence**: [which quantities vary: B(t), A(t), v(t), theta(t)]
- **Relevant parameters**: [B magnitude, loop dimensions, velocity, angular frequency]

期待結果: A clear identification of why the flux changes, what surface to integrate over, and which physical quantities carry the time dependence.

失敗時: If the source of changing flux is ambiguous (e.g., a deforming loop in a non-uniform field), decompose the problem into a sum of contributions: one from the field change at fixed geometry, and one from the geometry change in the instantaneous field. This decomposition is always valid.

ステップ2: Calculate Magnetic Flux Through the Relevant Surface

Compute the magnetic flux Phi_B = integral of B . dA over the surface S:

1. Uniform field, flat loop: Phi_B = B * A * cos(theta), where theta is the angle between B and the area normal vector n_hat. This is the most common textbook case.

2. Non-uniform field: Parameterize the surface S and evaluate the integral:

   • Choose coordinates aligned with the surface (e.g., polar for a circular loop)
   • Express B(r) at each point on the surface
   • Compute the dot product B . dA = B . n_hat dA
   • Integrate over the surface

3. Coupled coils (mutual inductance): For coil 2 linked to coil 1:

   • Compute B_1 (field from coil 1) at the location of coil 2
   • Integrate B_1 over the area of each turn of coil 2
   • Multiply by N_2 (number of turns in coil 2) for total flux linkage: Lambda_21 = N_2 * Phi_21
   • Mutual inductance: M = Lambda_21 / I_1

4. Self-inductance: For a single coil carrying current I:

   • Compute B inside the coil from the coil's own current
   • Integrate B over one turn's cross-section and multiply by N
   • Self-inductance: L = N * Phi / I = Lambda / I
   • Known results: solenoid L = mu_0 * n^2 * A * l; toroid L = mu_0 * N^2 * A / (2 pi R)

5. Time dependence: Express Phi_B(t) explicitly by substituting the time-varying quantities identified in Step 1.

## Flux Calculation
- **Flux expression**: Phi_B(t) = [formula]
- **Evaluation**: [analytic / numeric]
- **Flux linkage** (if multi-turn): Lambda = N * Phi_B = [formula]
- **Inductance** (if applicable): L = [value with units] or M = [value with units]

期待結果: An explicit expression for Phi_B(t) with correct units (Weber = T . m^2) and, if applicable, inductance values with units of Henry.

失敗時: If the flux integral cannot be evaluated analytically (e.g., non-uniform field over a non-trivial surface), use numerical quadrature. For mutual inductance of complex geometries, consider the Neumann formula: M = (mu_0 / 4 pi) * double_contour_integral of (dl_1 . dl_2) / |r_1 - r_2|.

ステップ3: Apply Faraday's Law for Induced EMF

Compute the induced EMF from the time derivative of the flux:

1. Faraday's law: EMF = -d(Lambda)/dt = -N * d(Phi_B)/dt. The negative sign encodes Lenz's law (opposition to the change).

2. Differentiation: Take the total time derivative of Phi_B(t):

   • If B = B(t) and A, theta are constant: EMF = -N * A * cos(theta) * dB/dt
   • If theta = omega * t (rotating coil in static B): EMF = N * B * A * omega * sin(omega * t)
   • If the area changes (e.g., sliding rail): EMF = -B * l * v (motional EMF, where l is the rail length and v the velocity)
   • For the general case: use the Leibniz integral rule to differentiate under the integral sign

3. Motional EMF (alternative derivation): For a conductor of length l moving with velocity v in field B:

   • The Lorentz force on charges in the conductor: F = q(v x B)
   • EMF = integral of (v x B) . dl along the conductor
   • This is equivalent to Faraday's law but can be more intuitive for moving conductors

4. Sign and magnitude check: The magnitude of EMF should be physically reasonable. For typical laboratory setups: mV to V range. For power generation: V to kV range.

## Induced EMF
- **EMF expression**: EMF(t) = [formula]
- **Peak EMF** (if AC): EMF_0 = [value with units]
- **RMS EMF** (if AC): EMF_rms = EMF_0 / sqrt(2) = [value]
- **Derivation method**: [Faraday's law / motional EMF / Leibniz rule]

期待結果: An explicit expression for EMF(t) with correct units (Volts) and physically reasonable magnitude.

失敗時: If the EMF has wrong units, trace back to the flux calculation -- a missing factor of area or an inconsistent unit system (e.g., mixing CGS and SI) is the most likely cause. If the EMF sign seems wrong, re-examine the surface normal orientation relative to the circuit loop direction (right-hand rule).

ステップ4: Determine Current Direction via Lenz's Law

Establish the direction of the induced current and its physical consequences:

1. Lenz's law statement: The induced current flows in the direction that opposes the change in magnetic flux that produced it. This is a consequence of energy conservation.

2. Application procedure:

   • Determine whether the flux through the loop is increasing or decreasing
   • If flux is increasing: induced current creates a B-field that opposes the increase (opposing the external field direction through the loop)
   • If flux is decreasing: induced current creates a B-field that supports the decreasing flux (same direction as the external field through the loop)
   • Use the right-hand rule to convert the required B-field direction into a current direction

3. Force consequences: The induced current in the presence of the external B-field experiences a force:

   • Eddy current braking: the force opposes the relative motion (always decelerating)
   • Magnetic levitation: the repulsive force supports weight when the geometry is appropriate
   • These forces are a direct manifestation of Lenz's law at the mechanical level

4. Qualitative verification: The induced effects should always resist the change. A falling magnet through a conducting tube falls slower than in free fall. A generator requires mechanical work input to produce electrical energy.

## Current Direction
- **Flux change**: [increasing / decreasing]
- **Induced B direction**: [opposing increase / supporting decrease]
- **Current direction**: [CW / CCW as viewed from specified direction]
- **Mechanical consequence**: [braking force / levitation / energy transfer]

期待結果: A clearly stated current direction that is consistent with Lenz's law, with the physical consequence (force, braking, energy transfer) identified.

失敗時: If the current direction seems to amplify the flux change rather than oppose it, the surface normal orientation or the right-hand rule application is reversed. Re-examine the loop orientation convention. A current that reinforces the flux change would violate energy conservation.

ステップ5: Solve Resulting Circuit Equation

Formulate and solve the circuit equation including the inductance:

1. RL circuit formation: When the induced EMF drives current through a circuit with resistance R and inductance L, Kirchhoff's voltage law gives:

   • Energizing (switch closes onto DC source V_0): V_0 = L dI/dt + R I
   • De-energizing (source removed, loop closed): 0 = L dI/dt + R I
   • General (time-varying EMF): EMF(t) = L dI/dt + R I

2. Solution of the first-order ODE:

   • Energizing: I(t) = (V_0 / R) * [1 - exp(-t / tau)], where tau = L / R is the time constant
   • De-energizing: I(t) = I_0 * exp(-t / tau)
   • AC drive EMF = EMF_0 sin(omega t): solve using phasor methods or particular + homogeneous solution
   • Transient duration: current reaches ~63% of final value after 1 tau, ~95% after 3 tau, ~99.3% after 5 tau

3. Energy analysis:

   • Energy stored in the inductor: U_L = (1/2) L I^2
   • Energy stored in the magnetic field per unit volume: u_B = B^2 / (2 mu_0) in vacuum, or u_B = (1/2) B . H in magnetic materials
   • Power dissipated in resistance: P_R = I^2 R
   • Energy conservation: rate of energy input = rate of energy storage + rate of dissipation

4. Mutual inductance coupling: For two coupled coils with mutual inductance M:

   • V_1 = L_1 dI_1/dt + M dI_2/dt + R_1 I_1
   • V_2 = M dI_1/dt + L_2 dI_2/dt + R_2 I_2
   • Coupling coefficient: k = M / sqrt(L_1 L_2), where 0 <= k <= 1
   • Solve the coupled ODEs simultaneously (matrix exponential or Laplace transform)

5. Steady-state and transient separation: For AC-driven circuits, decompose the solution into a transient (decaying exponential) and steady-state (sinusoidal at the drive frequency). Report impedance Z_L = j omega L and phase angle.

## Circuit Solution
- **Circuit type**: [RL energizing / de-energizing / AC driven / coupled coils]
- **Time constant**: tau = L/R = [value with units]
- **Current solution**: I(t) = [expression]
- **Energy stored**: U_L = [value at specified time]
- **Energy dissipated**: [total or rate]
- **Steady-state impedance** (if AC): Z_L = [value]

期待結果: A complete time-domain solution for the current with correct exponential time constants, energy balance verified, and physically reasonable magnitudes.

失敗時: If the current grows without bound, a sign error in the ODE setup is likely (the inductance term should oppose changes in current). If the time constant is unreasonably large or small, double-check the inductance calculation from Step 2 and the resistance value. Time constants for typical laboratory RL circuits range from microseconds to seconds.

バリデーション

• Source of changing flux is clearly identified (changing B, changing area, motional, combined)
• Magnetic flux integral is set up over the correct surface with proper orientation
• Flux has correct units (Weber = T . m^2)
• Inductance values (self or mutual) have correct units (Henry) and reasonable magnitude
• EMF has correct units (Volts) and physically reasonable magnitude
• EMF sign is consistent with Lenz's law (opposes the flux change)
• Current direction is determined by Lenz's law and verified with the right-hand rule
• RL circuit ODE is correctly set up with proper signs
• Time constant tau = L/R has correct units (seconds) and reasonable magnitude
• Energy balance is verified: input energy = stored energy + dissipated energy
• Limiting cases checked (t -> 0 for initial conditions, t -> infinity for steady state)

よくある落とし穴

• Wrong sign in Faraday's law: The EMF is EMF = -d(Lambda)/dt, not +d(Lambda)/dt. The negative sign is essential -- it encodes Lenz's law and energy conservation. Omitting it produces a current that amplifies the flux change, violating thermodynamics.
• Confusing flux and flux linkage: For a single-turn loop, Phi_B and Lambda are the same. For an N-turn coil, Lambda = N * Phi_B. Inductance is defined as L = Lambda / I, not L = Phi_B / I. Missing the factor of N produces inductance values that are N times too small.
• Surface normal inconsistency: The surface normal n_hat must be related to the loop circulation direction by the right-hand rule. Choosing them independently leads to sign errors in both the flux and the EMF.
• Ignoring back-EMF in RL circuits: When current changes in an inductor, the inductor generates a back-EMF that opposes the change. Omitting this term from Kirchhoff's voltage law makes the circuit equation algebraic instead of differential, missing the transient entirely.
• Assuming instantaneous current change: Current through an ideal inductor cannot change instantaneously (it would require infinite voltage). Initial conditions for RL transients must satisfy continuity of inductor current across switching events.
• Neglecting eddy currents in bulk conductors: Faraday's law applies to any closed path in a conductor, not just discrete wire loops. Time-varying fields in bulk conductors induce distributed eddy currents that produce heating (loss) and opposing fields (shielding). These are critical in transformer cores and must be minimized with lamination.

関連スキル

• analyze-magnetic-field -- compute the B-field from current distributions that serve as the flux source
• formulate-maxwell-equations -- generalize induction to the full Maxwell framework including displacement current
• design-electromagnetic-device -- apply induction principles to motors, generators, and transformers
• derive-theoretical-result -- derive analytic results for inductance, EMF, or transient solutions from first principles