AP Physics C E&M- 13.2 Electromagnetic Induction - Study Notes- New Syllabus
AP Physics C E&M- 13.2 Electromagnetic Induction – Study Notes
AP Physics C E&M- 13.2 Electromagnetic Induction – Study Notes – per latest Syllabus.
Key Concepts:
- Induced Electric Potential Difference from a Changing Magnetic Flux
- Lenz’s Law and the Direction of Induced emf
- Maxwell’s Third Equation: Faraday’s Law of Induction
Induced Electric Potential Difference from a Changing Magnetic Flux
A changing magnetic flux through a loop or coil induces an electromotive force (emf), which can drive a current if the loop is part of a closed circuit. This is described by Faraday’s Law of Induction.
Faraday’s Law:
\( \mathrm{\mathcal{E}_{ind} = – \dfrac{d\Phi_B}{dt}} \)
- \(\mathrm{\mathcal{E}_{ind}}\) = induced emf
- \(\mathrm{\Phi_B = \displaystyle \int \vec{B} \cdot d\vec{A}}\) = magnetic flux
- The negative sign represents Lenz’s Law: the induced emf always opposes the change in flux.
Special Cases:
Constant Area, Changing Magnetic Field:
\( \mathrm{\mathcal{E}_{ind} = -A \dfrac{dB_\perp}{dt}} \)
Constant Magnetic Field, Changing Area:
\( \mathrm{\mathcal{E}_{ind} = -B \dfrac{dA_\perp}{dt}} \)
Solenoid with \(N\) Loops:
\( \mathrm{\mathcal{E}_{total} = -N \dfrac{d\Phi_B}{dt}} \)
Key Ideas:
- The induced emf is proportional to the rate of change of magnetic flux, not the flux itself.
- Induction can occur due to:
- Changing magnetic field strength.
- Changing orientation (angle) of the loop relative to the field.
- Changing area of the loop.
- The direction of induced current can be found using Lenz’s Law (opposes flux change).
Example:
A circular loop of radius \(0.10 \, m\) lies in a uniform magnetic field directed perpendicular to the plane of the loop. The field decreases uniformly from \(0.50 \, T\) to \(0 \, T\) in \(0.20 \, s\). Find the induced emf in the loop.
▶️ Answer/Explanation
Step 1: Area of loop: \( \mathrm{A = \pi r^2 = \pi (0.10)^2 = 0.0314 \, m^2} \).
Step 2: Change in flux: \( \mathrm{\Delta \Phi_B = A \Delta B = (0.0314)(0.50 – 0)} = 0.0157 \, Wb \).
Step 3: Induced emf: \( \mathrm{\mathcal{E}_{ind} = – \dfrac{\Delta \Phi_B}{\Delta t} = – \dfrac{0.0157}{0.20}} \).
Step 4: \( \mathrm{\mathcal{E}_{ind} = -0.0785 \, V} \). (Magnitude: \(0.079 \, V\)).
Final Answer: The induced emf is \(0.079 \, V\). The negative sign indicates that the induced current will oppose the decreasing magnetic flux (Lenz’s Law).
Lenz’s Law and the Direction of Induced emf
Lenz’s law describes the direction of the induced emf and current caused by a changing magnetic flux. The induced emf always produces a current whose magnetic field opposes the change in the original magnetic flux.
Mathematical Statement (Faraday’s Law with Lenz’s Sign):
\( \mathrm{\mathcal{E}_{ind} = – \dfrac{d\Phi_B}{dt}} \)
- The negative sign explicitly represents Lenz’s Law.
- \(\mathrm{\Phi_B = \displaystyle \int \vec{B} \cdot d\vec{A}}\) = magnetic flux through the loop.
Key Ideas:
- If magnetic flux through a loop is increasing, the induced current produces a magnetic field opposing the increase.
- If magnetic flux is decreasing, the induced current produces a magnetic field that reinforces the flux (opposes the decrease).
- This is a consequence of the conservation of energy: induced currents resist the change in flux.
Right-Hand Rule Application:
- Point the thumb of the right hand in the direction of the magnetic field produced by the induced current (the direction that opposes the flux change).
- The curl of the fingers shows the direction of the induced current.
- This connects the change in flux, the induced emf, and the resulting current loop.
Key Idea: Lenz’s Law ensures that induced currents and fields always oppose changes in flux, preventing violation of energy conservation. The right-hand rule provides the physical orientation of the induced current.
Example:
A circular conducting loop lies in a region with a magnetic field pointing into the page. The magnitude of the magnetic field is decreasing with time. What is the direction of the induced current?
▶️ Answer/Explanation
Step 1: The flux into the page is decreasing.
Step 2: By Lenz’s law, the induced current must create a magnetic field into the page to oppose this decrease.
Step 3: Use the right-hand rule: curl fingers in the direction of current so that the thumb points into the page.
Step 4: The current must circulate in the clockwise direction.
Final Answer: The induced current flows clockwise around the loop.
Maxwell’s Third Equation: Faraday’s Law of Induction
Faraday’s law of induction states that a changing magnetic flux induces an electric field. This induced electric field can drive a current in a conducting loop or exist as a nonconservative electric field in space.
Differential Form:
\( \mathrm{\nabla \times \vec{E} = – \dfrac{\partial \vec{B}}{\partial t}} \)
- The curl of the electric field is related to the time rate of change of the magnetic field.
- This shows that time-varying magnetic fields produce circulating electric fields.
Integral Form:
\( \mathrm{\oint_{\partial A} \vec{E} \cdot d\vec{s} = – \dfrac{d}{dt} \displaystyle \int_A \vec{B} \cdot d\vec{A}} \)
- The line integral of the electric field around a closed loop equals the negative time derivative of the magnetic flux through the loop.
- This is consistent with the induced emf expression: \( \mathrm{\mathcal{E} = – \dfrac{d\Phi_B}{dt}} \).
Key Ideas:
- Unlike electrostatic fields, induced electric fields are nonconservative (their line integral around a closed loop is not zero).
- Maxwell’s equations show that a changing magnetic field induces an electric field, and a changing electric field induces a magnetic field (via Ampère’s law with Maxwell’s addition).
- Together, these laws explain how self-sustaining electric and magnetic fields propagate as electromagnetic waves.
- Electromagnetic waves travel in free space at a constant speed:
\( \mathrm{c = \dfrac{1}{\sqrt{\mu_0 \varepsilon_0}} \approx 3.0 \times 10^8 \, m/s} \)
Example:
A magnetic field through a circular loop of radius \(0.05 \, m\) increases uniformly at a rate of \(0.20 \, T/s\). Find the magnitude of the induced electric field at the edge of the loop.
▶️ Answer/Explanation
Step 1: Use Faraday’s law in integral form: \( \mathrm{\oint \vec{E} \cdot d\vec{s} = – \dfrac{d\Phi_B}{dt}} \).
Step 2: For circular symmetry: \( \mathrm{E (2\pi r) = – A \dfrac{dB}{dt}} \), where \( A = \pi r^2 \).
Step 3: Solve for \(E\): \( \mathrm{E = – \dfrac{r}{2} \dfrac{dB}{dt}} \).
Step 4: Substitute values: \( \mathrm{E = \dfrac{0.05}{2}(0.20) = 0.005 \, V/m} \).
Final Answer: The induced electric field at the loop’s edge is \( \mathrm{5.0 \times 10^{-3} \, V/m} \).