29.10:
Ampere’s Law: Problem-Solving
Consider an infinite straight, thick conductor carrying a uniform current. What is the magnetic field inside and outside the conductor?
Here, the current distribution possesses a cylindrical symmetry, and the circular magnetic field lines are concentric with the conductor axis.
To estimate the field inside the conductor, consider an Amperian loop with a radius less than the conductor radius.
The line integral of the magnetic field along this loop equals the magnetic field times the circumference. The current enclosed by the loop equals the product of current density and the enclosed area.
Thus, using Ampere's Law, the magnetic field inside the conductor is obtained.
For an Amperian loop with a radius greater than the conductor radius, the current enclosed by the loop equals the current flowing through it.
Again, applying Ampere's Law, the magnetic field outside the conductor is inversely proportional to the loop radius.
The magnitude of the magnetic field inside the conductor increases linearly till it reaches a maximum value at its radius. Outside the conductor, the field magnitude drops inversely with distance.
29.10:
Ampere’s Law: Problem-Solving
Ampere's law states that for any closed looped path, the line integral of the magnetic field along the path equals the vacuum permeability times the current enclosed in the loop. If the fingers of the right hand curl along the direction of the integration path, the current in the direction of the thumb is considered positive. The current opposite to the thumb direction is considered negative.
Specific steps need to be considered while calculating the symmetric magnetic field distribution using Ampere's Law.
- The symmetry of the current is identified. For a non-symmetric current distribution, the magnetic field can be calculated using Biot Savart's Law instead of Ampere's Law.
- A symmetric integration path is chosen where the magnetic field is either constant or zero. Ideally, this Amperian path should be tangential or perpendicular to the magnetic field along the path.
- When a constant magnetic field is tangent to all or some portion of the Amperian path, the line integral of the magnetic field reduces to the product of the constant magnetic field times the length of the loop for that portion. The line integral is zero for regions where the magnetic field is perpendicular to the path, or its magnitude is zero.
- The current enclosed by the integration path is calculated by summing up the individual currents passing through the path. The right-hand rule gives the direction of the current. If the curl of the fingers follows the direction of the path integration, then the thumb points in the direction of the positive current.
- If the magnetic field is tangential to the path and the current enclosed is positive, then the magnetic field's direction follows the integration direction. If the enclosed current is negative, the direction of the magnetic field is opposite to the direction of integration.
- Finally, the line integral of the magnetic field is equated with the current enclosed to get the magnetic field.
Suggested Reading
- Young, H. D., and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson. pp. 938.
- OpenStax. (2019). University Physics Vol. 2. [Web version]. pp.551 Retrieved fromhttps://openstax.org/books/college-physics/pages/22-9-magnetic-fields-produced-by-currents-amperes-law