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Lenz's Law: Induction in a Circular Loop
Developed by Stephen Robinson  Published October 19, 2017
The study of physics is very heavily dependent on both mathematical and conceptual understandings. Many beginning students possess these abilities independently but struggle to combine them or see how they are related. For example, to a trained physicist, a simple negative sign can reveal important ideas about direction, behavior, or whether a fundamental quantity is conserved. For firstyear students, however, that negative sign may just be seen as an unimportant artifact or something to add to an answer when the problem is complete. In reality, the “math” and the “concept” are simply two different expressions of the same underlying reality; the sooner that students see this, the sooner they will master physics.
Lenz's law states that an induced current flows in a direction that creates a magnetic field that opposes the original changing magnetic flux. Most students wrestle with this conceptual language. This exercise is designed to help students visualize Lenz's law in three dimensions plus time. They will do this in conjunction with calculations of Faraday's law, the mathematical description of the concept they will model. By simultaneously working with both, these relationships will become clearer, specifically with regard to how currents and magnetic fields are related through the use of derivatives and negative signs.
Subject Area  Electricity & Magnetism 

Level  First Year 
Available Implementation  VPython 
Learning Objectives 
Students who complete this Exercise Set will be able to:
* calculate the magnitude of an induced magnetic field in a loop (**Exercise 1**).
* display that magnetic field in three dimensions alongside an applied magnetic field and induced current (**Exercise 2**).
* make connections between the quantitative (i.e., equationbased) and qualitative (i.e., visualizationbased) representations of Lenz's law. (**Exercise 3**).

These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).
####Exercise 1
An external applied magnetic field (uniform over space, sinusoidal in time) exists perpendicular to the plane of a single circular loop of wire. Create a program that asks a user for the field amplitude (T), frequency (Hz), loop radius (m), and loop resistance (ohms). The program should then output the induced magnetic field amplitude (T) in the center of the loop.
The following shows a stepbystep solution that should be attempted by hand before programming.
* applied magnetic field: $B_\mathrm{{ext}}(t)=B_0 \mathrm{sin}(2\pi ft)$
* angular frequency: $w = 2\pi f$
* magnetic flux: $\Phi_B= \int \vec B\bullet d\vec A = B_0\pi r^2\mathrm{sin}(\omega t)$
* induced emf: $\mathcal{E}(t) = \displaystyle\frac{d\Phi_B}{dt} = B_0\pi r^{2}\omega \mathrm{cos}(\omega t) = \mathcal{E_\mathrm{max}}\mathrm{cos}(\omega t)$
* induced current amplitude: $I_\mathrm{{max}} = \displaystyle\frac{\mathcal{E}_\mathrm{{max}}}{R}$
* induced current: $I(t)=I_\mathrm{{max}}\mathrm{cos}(\omega t)$
* induced magnetic field amplitude in the center of the loop (via the BiotSavart Law): $B_\mathrm{{max}} = \displaystyle\frac{\mu_0I_\mathrm{{max}}}{2r}\rightarrow B_\mathrm{{ind}}(t)=B_\mathrm{{max}}\mathrm{cos}(\omega t)$
With $B_0$ = 100 mT, $f$ = 60 Hz, $r$ = 5 cm, and $R$ = 0.5 $\Omega$, this gives $B_\mathrm{{max}} = 7.44\times10^{6}$ T in the center of the loop.
####Exercise 2
Display the directions of 1) the external magnetic field, 2) the induced magnetic field, and 3) the current in the loop over time. Be sure to distinguish the arrows with labels. The directions of the current and magnetic field are oriented in accordance with Lenz's law as described above.
####Exercise 3
Answer the following conceptual questions through calculations (e.g., by hand or spreadsheet) and/or programming.
* What is true about the external magnetic field (and its time rate of change) when the induced magnetic field is at an extreme, and vice versa? How do you know visually, and how is it seen in the relevant equations for each?
* Mathematically, what causes the external and induced magnetic fields to be 90° out of phase? Would this happen for any function of the external field, or is it specific to this particular function? Similarly, what causes the induced current and induced magnetic field to be in phase, and is that functiondependent?
* If the magnitude of the external magnetic field were linearly increasing in the upward direction, how would the induced magnetic field behave (in both magnitude and direction)?
* If the magnitude of the external magnetic field were exponentially increasing in the upward direction, how would the induced magnetic field behave (in both magnitude and direction)?
* If the external magnetic field were constant (and nonzero), how would you expect the induced magnetic field to behave? What if the ring were moving or began to tilt?
* What happens to the induced magnetic field when the frequency is increased? Why?
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Credits and Licensing
Stephen Robinson, "Lenz's Law: Induction in a Circular Loop," Published in the PICUP Collection, October 2017.
The instructor materials are ©2017 Stephen Robinson.
The exercises are released under a Creative Commons AttributionNonCommercialShareAlike 4.0 license