Section 4.2 | Electrodynamics

Motional Induction in Rectangular Geometries

In the study of electromagnetism, few scenarios provide as clear a window into the interplay between mechanics and field theory as a conductor moving through a magnetic field. When a rectangular loop enters a region of uniform magnetic flux density \( \vec{B} \), a series of predictable yet profound responses occur.

I. The Geometry of the System

Consider a coordinate system where a uniform magnetic field \( \vec{B} \) is directed into the page. A rectangular loop lies in the \( xy \)-plane and moves with a constant velocity \( \vec{v} \) in the \( +x \) direction.

Loop

\( \ell \)

\( w \)

v

Figure 1: Loop geometry approaching a uniform \( B \)-field.

The total magnetic flux \( \Phi_B \) through the loop is: \( \Phi_B = B A_{in} \). As the loop moves, the immersed area \( A_{in} \) changes, inducing an EMF according to Faraday's Law.

Checkpoint 1.1

If the magnetic field were not perpendicular to the loop, how would the flux equation be modified?

PAGE 01

Phase I: Entering the Field

As the leading edge enters the field, the immersed area increases linearly with time: \( A(t) = \ell v t \). Consequently, the flux increases:

\[ \Phi_B(t) = B \ell v t \]

The Induced EMF

Faraday's Law states that the induced EMF \( \varepsilon \) is the negative rate of change of magnetic flux:

\[ \varepsilon = - \frac{d\Phi_B}{dt} = - B \ell v \]

Lenz’s Law and Current

Flux into the page is increasing. The loop opposes this change by creating an outward field, resulting in a counter-clockwise (CCW) current: \( I = B\ell v / R \).

The Magnetic Drag Force

Using the Lorentz force law \( \vec{F} = I\vec{\ell} \times \vec{B} \), we find the force on the leading edge points to the left, opposing the loop's velocity.

Conclusion

Entering the field creates a magnetic drag force that resists the physical motion of the loop.

Reflect

Explain why the forces on the top and bottom segments of the loop do not contribute to the net horizontal force.

PAGE 02

Phase II: Fully Immersed

When the loop is entirely within the field, the area inside the field is constant: \( A = \ell \cdot w \).

The Vanishing EMF

Because the magnetic field is uniform and the area is not changing, the flux is constant. The derivative of a constant is zero:

\[ \frac{d\Phi_B}{dt} = 0 \implies \varepsilon = 0 \]

With no induced EMF, no current flows (\( I = 0 \)), and there is no net magnetic force.

Theoretical Insight

Both vertical segments generate individual motional EMFs. However, because they are connected in series but pointing in "opposite" directions relative to the loop's circulation, they cancel each other out perfectly.

Analyze

If the magnetic field were not uniform, would the EMF still be zero in this phase? Justify your answer.

PAGE 03

Phase III: Exiting the Field

As the loop exits, the flux into the page decreases. Faraday’s Law yields a positive EMF:

\[ \varepsilon = - \frac{d\Phi_B}{dt} = + B \ell v \]

Lenz’s Law Response

To replace the lost flux, the loop induces a clockwise (CW) current. This current creates its own magnetic field directed into the page.

The Braking Effect

Only the trailing edge remains in the field. In this segment, current flows upward, and the magnetic field points in. The resulting force \( \vec{F}_B \) is directed to the left.

Conservation Principle

The magnetic force always opposes the change—in this case, it resists the loop leaving the field. This is the mechanical manifestation of Lenz's Law.

Verify

Using the RHR, confirm the direction of force for the trailing edge during the exit phase.

PAGE 04

IV. Summary of Findings

Property	Entering	Inside	Exiting
Flux	Increasing	Constant	Decreasing
Induced EMF	Negative (-)	Zero (0)	Positive (+)
Current Direction	CCW	None	CW
Magnetic Force	LEFT	Zero	LEFT

Power Dissipation

The mechanical power required to maintain motion equals the electrical heat dissipated:

\[ P = \frac{B^2 \ell^2 v^2}{R} \]

Synthesis Challenge

Calculate the total energy dissipated in the resistor from the moment the loop starts to enter until it has completely left the field.

Key Takeaway: Induction creates a mechanical resistance that preserves the symmetry of energy.

PAGE 05

V. Graphical Signatures of Induction

Magnetic Flux (\( \Phi_B \)) vs. Time

Start Entry Full Entry Start Exit Full Exit

\( \Phi_{max} \)

0

Figure 2: The magnetic flux through the loop rises linearly during penetration, remains constant while fully immersed, and falls linearly during the exit.

Induced EMF (\( \varepsilon \)) vs. Time

\( +B\ell v \)

\( -B\ell v \)

Entry Inside Exit

Figure 3: The induced EMF is a derivative of flux. Since flux is linear in transition phases, EMF is a constant non-zero value during those periods.

PAGE 06

Induction Treatise Reading

Induction Treatise Reading

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