Faraday’s Law
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Statement: The induced EMF in a closed loop equals the negative rate of change of magnetic flux through the loop.
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Mathematical Form:
\[\boxed{\mathcal{E} = -\frac{d}{dt} \Phi_B} \quad \text{where} \quad \Phi_B = \int \mathbf{B} \cdot d\mathbf{A}\] -
Alternative Forms:
\[\begin{aligned} \mathcal{E} &= -\frac{d}{dt} \int \mathbf{B} \cdot d\mathbf{A} \\ \oint \mathbf{E} \cdot d\mathbf{l} &= -\frac{d}{dt} \int \mathbf{B} \cdot d\mathbf{A} \end{aligned}\] -
Lenz’s Law: The negative sign indicates that induced current opposes the change in flux.
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Magnetic Flux (\(\Phi_B\)):
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Unit: Weber (Wb) = T m2
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\(\Phi_B = B \cdot A \cdot \cos\theta\) (for uniform field)
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\(\theta\): Angle between \(\mathbf{B}\) and area normal
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Induced EMF (\(\mathcal{E}\)):
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Unit: Volt (V)
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Direction determined by Lenz’s Law
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Three ways to change flux:
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Change magnetic field strength (\(B\))
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Change loop area (\(A\))
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Change orientation (\(\theta\))
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Moving Conductor in Magnetic Field:
where \(l\) = length, \(v\) = velocity, \(\theta\) = angle between \(\mathbf{v}\) and \(\mathbf{B}\)\[\mathcal{E} = B \cdot l \cdot v \cdot \sin\theta\] -
Rotating Coil in Magnetic Field:
where \(N\) = number of turns, \(\omega\) = angular frequency\[\mathcal{E} = NBA\omega \sin(\omega t)\] -
Transformers: \(\frac{\mathcal{E}_1}{\mathcal{E}_2} = \frac{N_1}{N_2}\)
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GATE Tip: Always check the direction of induced current using right-hand rule!
Lorentz Force
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Definition: Total force on a charged particle in electric and magnetic fields.
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Complete Formula:
\[\boxed{\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})}\] -
Component Analysis:
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Electric Force: \(\mathbf{F}_E = q\mathbf{E}\) (parallel to \(\mathbf{E}\))
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Magnetic Force: \(\mathbf{F}_B = q\mathbf{v} \times \mathbf{B}\) (perpendicular to both \(\mathbf{v}\) and \(\mathbf{B}\))
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Magnitude of Magnetic Force:
where \(\theta\) is angle between \(\mathbf{v}\) and \(\mathbf{B}\)\[|\mathbf{F}_B| = |q|vB\sin\theta\]
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Cyclotron Motion:
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Radius: \(r = \frac{mv}{qB}\)
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Frequency: \(f = \frac{qB}{2\pi m}\) (independent of \(v\)!)
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Period: \(T = \frac{2\pi m}{qB}\)
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Velocity Selector: \(v = \frac{E}{B}\) (when \(\mathbf{E} \perp \mathbf{B}\))
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Hall Effect:
where \(n\) = charge density, \(t\) = thickness\[V_H = \frac{IB}{nqt} \quad \text{(Hall Voltage)}\] -
GATE Focus: Direction problems using right-hand rule, cyclotron calculations
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Direction Rules:
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Right-hand rule: Point fingers in direction of \(\mathbf{v}\), curl towards \(\mathbf{B}\), thumb shows \(\mathbf{F}\) for positive charge
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For negative charge: Reverse the direction
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Special Cases:
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\(\mathbf{v} \parallel \mathbf{B}\): No magnetic force (\(\sin\theta = 0\))
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\(\mathbf{v} \perp \mathbf{B}\): Maximum force (\(\sin\theta = 1\))
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Stationary charge: Only electric force
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Energy Considerations:
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Magnetic force does NO work (always \(\perp\) to motion)
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Electric force can do work
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Inductance
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Definition: Property of a conductor/circuit to oppose changes in current through induced EMF.
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Physical Origin: Changing current creates changing magnetic field, which induces EMF (Faraday’s Law)
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Unit: Henry (H)
\[1 \text{ Henry} = 1 \frac{\text{Weber}}{\text{Ampere}} = 1 \frac{\text{Volt·second}}{\text{Ampere}}\] -
Energy Storage:
\[U = \frac{1}{2}LI^2 \quad \text{(Energy stored in inductor)}\] -
Key Insight: Inductance depends only on geometry and material properties, not on current!
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Self-Inductance (\(L\)):
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EMF induced in same circuit due to its own changing current
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\(\mathcal{E} = -L \frac{dI}{dt}\)
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Always opposes change in current
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Mutual Inductance (\(M\)):
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EMF induced in one circuit due to changing current in another
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\(\mathcal{E}_2 = -M \frac{dI_1}{dt}\)
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Basis for transformers
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Relationship:
\[M_{12} = M_{21} = M \quad \text{(Reciprocity)}\] -
GATE Note: Both self and mutual inductance can be positive or negative depending on winding direction!
Self and Mutual Inductance
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General Definition:
where \(N\) = number of turns\[L = \frac{\Phi_B}{I} = \frac{N\Phi_B}{I}\] -
Induced EMF:
\[\mathcal{E} = -L \frac{dI}{dt}\] -
From Magnetic Field Energy:
\[L = \frac{2U_B}{I^2} \quad \text{where} \quad U_B = \frac{B^2}{2\mu_0} \times \text{Volume}\] -
Important: Self-inductance is always positive in our sign convention
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Solenoid (Long):
where \(n = N/l\) (turns per unit length)\[\boxed{L = \mu_0 n^2 A l = \frac{\mu_0 N^2 A}{l}}\] -
Toroidal Coil:
where \(r\) = mean radius of torus\[L = \frac{\mu_0 N^2 A}{2\pi r}\] -
Single-turn Circular Loop:
where \(R\) = loop radius, \(a\) = wire radius\[L = \mu_0 R \left[\ln\left(\frac{8R}{a}\right) - 2\right]\] -
GATE Focus: Solenoid formula is most frequently asked!
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Definition:
\[M_{21} = \frac{\Phi_{21}}{I_1} = \frac{N_2\Phi_{21}}{I_1}\] -
Induced EMF:
\[\mathcal{E}_2 = -M \frac{dI_1}{dt}\] -
Coupling Coefficient:
\[k = \frac{M}{\sqrt{L_1 L_2}} \quad \text{where} \quad 0 \leq k \leq 1\]-
\(k = 0\): No coupling
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\(k = 1\): Perfect coupling
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Maximum Mutual Inductance:
\[M_{\text{max}} = \sqrt{L_1 L_2}\]
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Coaxial Solenoids:
\[M = \frac{\mu_0 N_1 N_2 A}{l} \quad \text{(inner solenoid area $A$)}\] -
Two Parallel Coaxial Circular Loops:
where \(K(k)\) and \(E(k)\) are elliptic integrals\[M = \frac{\mu_0 \sqrt{R_1 R_2}}{2} \left[(2-k^2)K(k) - 2E(k)\right]\] -
Transformer (Ideal):
\[M = k\sqrt{L_1 L_2} \quad \text{with} \quad k \approx 1\] -
GATE Tip: For most problems, use coaxial solenoid formula!
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Series Connection:
\[L_{\text{total}} = L_1 + L_2 + 2M \quad \text{(aiding)}\]\[L_{\text{total}} = L_1 + L_2 - 2M \quad \text{(opposing)}\] -
Parallel Connection:
\[\frac{1}{L_{\text{total}}} = \frac{1}{L_1} + \frac{1}{L_2} \quad \text{(when $M = 0$)}\] -
RL Circuit Time Constant:
\[\tau = \frac{L}{R} \quad \text{and} \quad I(t) = I_0\left(1 - e^{-t/\tau}\right)\] -
Important: Sign of mutual inductance depends on relative winding directions!
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For Faraday’s Law Problems:
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Identify what’s changing (B, A, or \(\theta\))
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Apply \(\mathcal{E} = -\frac{d\Phi_B}{dt}\)
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Use Lenz’s Law for direction
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For Lorentz Force Problems:
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Identify all fields present
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Apply right-hand rule for direction
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Calculate magnitude using \(F = qvB\sin\theta\)
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For Inductance Problems:
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Identify geometry (solenoid, toroidal, etc.)
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Apply appropriate formula
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Check units and reasonableness
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Faraday’s Law: \(\mathcal{E} = -\frac{d\Phi_B}{dt}\)
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Lorentz Force: \(\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})\)
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Cyclotron Frequency: \(f = \frac{qB}{2\pi m}\)
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Solenoid Self-Inductance: \(L = \frac{\mu_0 N^2 A}{l}\)
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Mutual Inductance: \(M = \frac{\mu_0 N_1 N_2 A}{l}\) (coaxial solenoids)
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Coupling Coefficient: \(k = \frac{M}{\sqrt{L_1 L_2}}\)
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Energy in Inductor: \(U = \frac{1}{2}LI^2\)
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RL Time Constant: \(\tau = \frac{L}{R}\)
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Master the Basics:
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Faraday’s Law: Focus on flux change calculations
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Lorentz Force: Practice direction problems
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Inductance: Memorize solenoid formulas
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Common GATE Topics:
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EMF in moving conductors
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Cyclotron motion calculations
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Transformer principles
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Mutual inductance of coaxial coils
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Problem-Solving Tips:
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Always check units in final answers
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Use symmetry to simplify calculations
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Draw clear diagrams for direction problems
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