## Ampere's Law

The integral form of Ampere's law is used to find the magnetic field in situations with high symmetry. Clean, coherent symmetry arguments are fundamental to the use of Ampere's law which can be developed by using Proof by Contradiction. There must be a sufficient amount of symmetry in the current distribution so that the field can be pulled out of the flux integral. In order to do this, the symmetry of the current distribution is used to make assumptions about the components and dependence of the field. Proof by Contradiction is used to justify these assumptions and is used by assuming the opposite of what one wants to show is true and demonstrates a contradiction.

Ampere's law, in integral form, relates the magnetic field and current by $\oint{\vec{B}\cdot d\vec{l}} =\mu_0 I$. This means that the circulation of the magnetic field through a closed curve is directly related to the current enclosed by the curve. In order to use Ampere's law, students should understand how to calculate enclosed current from current densities and be able to integrate line integrals over a closed curve.

The Gauss's Law activity can be used to introduce symmetry arguments by Proof by Contradiction where students calculate the electric field due to a highly symmetric charge distribution using Gauss's law. If students are already able to provide clean, coherent symmetry arguments, this activity can be skipped because it does not directly pertain to Ampere's law. This activity is very similarly structured to the Ampere's Law activity and can serve as an introduction to this sequence.

### Activities

• Acting Out Current Density (Estimated time: 10 minutes): This kinesthetic activity prompts many discussions about the definitions of various current densities by having students each represent a charge and move as a class to demonstrate various current densities. Students learn to distinguish between different dimensions of current densities, constant and uniform currents, and current as a flux. Through this activity, students can begin to understand the total current, which is used in Ampere's law, as the flux of the current density.
• Ampere's Law (Estimated time: 45 minutes): A small group activity which utilizes similar skills as those developed in the Gauss's Law activity by extending symmetry arguments using Proof by Contradiction in Ampere's law with highly symmetric current densities. Students practice finding the magnetic field due to a current density by finding the total current and making explicit symmetry arguments in the application of Ampere's law.

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