32-2 Induced Magnetic Fields

32-2 Induced Magnetic Fields#

Prompts

  • Explain how a changing electric flux induces a magnetic field. What is the symmetry with Faraday’s law, and why does Maxwell’s law have no minus sign?

  • What is the capacitor paradox? Ampère’s law says current produces a magnetic field, but between the plates of a charging capacitor there is no conduction current. Why does Ampère’s law alone give contradictory results for different surface choices, and what does Maxwell add to fix it?

  • To apply the Ampere–Maxwell law, I need to pick a surface bounded by my loop. The law involves both current through that surface and the rate of change of electric flux through it. Does it matter which surface I choose? Walk me through why the result is the same.

Lecture Notes#

Overview#

  • Faraday (Chap 30): Changing \(\vec{B}\) → induces \(\vec{E}\)

  • Maxwell: Changing \(\vec{E}\) → induces \(\vec{B}\)

  • Same structure: Both relate circulation of the induced field to the rate of change of flux

Physical intuition

Electric and magnetic fields are coupled: a changing field of one kind creates the other. This symmetry is central to electromagnetic waves and Maxwell’s equations.

Maxwell’s law of induction#

(207)#\[ \oint_C \vec{B} \cdot d\vec{s} = \mu_0 \varepsilon_0 \frac{d\Phi_E}{dt}\bigg\vert_\Sigma \]

  • \(C\): Closed loop — the curve along which we integrate \(\vec{B} \cdot d\vec{s}\)

  • \(\Sigma\): Any open surface bounded by \(C\) (s.t. \(\partial \Sigma = C\)). \(\Phi_E\vert_\Sigma = \int_\Sigma \vec{E} \cdot d\vec{A}\) is the flux through \(\Sigma\).

  • Statement: A changing electric flux through \(\Sigma\) induces a magnetic field around \(C\)

  • Direction: Right-hand rule with \(d\Phi_E/dt\) (no minus sign, unlike Faraday)

Sign convention

Faraday’s law has a minus sign (Lenz’s law: induced \(\vec{E}\) opposes the change).

(208)#\[ \oint_C \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}\bigg\vert_\Sigma \]

Maxwell’s law has no minus sign (see Eq. (207)). The opposite sign conventions are crucial: they allow electromagnetic waves to propagate in an oscillatory manner through a negative feedback loop.

Charging capacitor: induced \(\vec{B}\) from changing \(\vec{E}\)#

  • Setup: Parallel-plate capacitor being charged by constant current \(i\)

  • Loop \(C\): Circle of radius \(r\) between the plates, concentric with the plates. The same loop \(C\) can bound different surfaces.

Two choices of surface (both have \(\partial \Sigma = C\)):

Surface

Path

\(i_{\text{enc}}\)

\(d\Phi_E/dt\)

\(\Sigma_{\text{wire}}\)

Surface that cuts the wire

\(\neq 0\)

\(\sim 0\)

\(\Sigma_{\text{gap}}\)

Disk through the gap (avoids the wire)

\(0\)

\(\neq 0\) (growing \(\vec{E}\))
(a) (a)
(b) (b)

Fig. 20 Two choices of surface \(\Sigma\) bounded by loop \(C\): (a) \(\Sigma_{\text{wire}}\) (cuts wire) — Surface cutting the wire, \(i_{\text{enc}} = i\), conduction current enclosed. (b) \(\Sigma_{\text{gap}}\) (avoids wire) — Surface through the gap, \(i_{\text{enc}} = 0\), \(d\Phi_E/dt \neq 0\).#

  • Result: \(\vec{B}\) is induced around \(C\) (circular field lines, like a current)

The capacitor paradox

Paradox (Ampere’s law alone): For the same loop \(C\), \(\Sigma_{\text{gap}}\) gives \(i_{\text{enc}} = 0\) (no wire pierces it) while \(\Sigma_{\text{wire}}\) gives \(i_{\text{enc}} = i\). Ampere’s law \(\oint_C \vec{B} \cdot d\vec{s} = \mu_0 i_{\text{enc}}\) would give different values for \(\oint_C \vec{B} \cdot d\vec{s}\) — a contradiction, since the left-hand side is fixed by the actual \(\vec{B}\) field.

Resolution (Ampere–Maxwell law): On \(\Sigma_{\text{gap}}\), \(i_{\text{enc}} = 0\) but \(\mu_0\varepsilon_0\,d\Phi_E/dt\) is non-zero and equals \(\mu_0 i\). On \(\Sigma_{\text{wire}}\), both terms contribute. Both surfaces yield the same \(\oint_C \vec{B} \cdot d\vec{s}\).

Physical intuition: continuity of current

Between the plates there is no conduction current, but \(\vec{E}\) is changing. Maxwell’s term \(\mu_0\varepsilon_0\,d\Phi_E/dt\) acts like a “displacement current” and produces \(\vec{B}\) the same way a real current would. This closes the gap in Ampere’s law.

Ampere’s law vs Maxwell’s law#

Ampere’s law

Maxwell’s law

\(\oint_C \vec{B} \cdot d\vec{s} = \mu_0 i_{\text{enc}}\vert_\Sigma\)

\(\oint_C \vec{B} \cdot d\vec{s} = \mu_0\varepsilon_0 \frac{d\Phi_E}{dt}\big\vert_\Sigma\)

Current through \(\Sigma\) produces \(\vec{B}\)

Changing \(\Phi_E\) through \(\Sigma\) produces \(\vec{B}\)

Steady currents

Capacitors, time-varying fields

Ampere–Maxwell law (combined)#

(209)#\[ \oint_C \vec{B} \cdot d\vec{s} = \mu_0\varepsilon_0 \frac{d\Phi_E}{dt}\bigg\vert_\Sigma + \mu_0 i_{\text{enc}}\big\vert_\Sigma \]

  • \(C\): Closed loop; \(\Sigma\): open surface with \(\partial \Sigma = C\). Both \(\Phi_E\) and \(i_{\text{enc}}\) are computed on the same \(\Sigma\).

  • Both sources: \(\vec{B}\) from conduction current through \(\Sigma\) and from changing \(\Phi_E\) through \(\Sigma\)

  • Wire with steady current: \(d\Phi_E/dt\vert_\Sigma = 0\) → Ampere’s law only

  • Capacitor gap (no current through \(\Sigma\)): \(i_{\text{enc}}\vert_\Sigma = 0\) → Maxwell’s law only

General method

For any closed loop \(C\):

  1. Choose an open surface \(\Sigma\) with \(\partial \Sigma = C\).

  2. Compute \(i_{\text{enc}}\vert_\Sigma\) (current through \(\Sigma\)) and \(\Phi_E\vert_\Sigma = \int_\Sigma \vec{E} \cdot d\vec{A}\).

  3. Add both contributions.

The result is independent of which \(\Sigma\) you choose. Use symmetry (e.g., circular loops) to simplify.

Summary#

  • Changing \(\vec{E}\) induces \(\vec{B}\) (Maxwell’s law)

  • \(\oint_C \vec{B} \cdot d\vec{s} = \mu_0\varepsilon_0\,d\Phi_E/dt\vert_\Sigma\) (no minus sign); \(\Sigma\) is any open surface with \(\partial \Sigma = C\)

  • Charging capacitor: \(\vec{B}\) between plates from changing \(\vec{E}\) only

  • Ampere–Maxwell law: \(\vec{B}\) from current and from \(d\Phi_E/dt\)