Name: choosing an amperian loop for magnetic fields
Uploaded: 2026-07-09T14:42:43.904Z
Duration: 1 min 31 s
Channel: JoVE

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Motion and Stability- Forces and Interactions

PS.PSc.6.4

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&#x27;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&#x27;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.

ampere's law: problem-solving

Ampere&#x27;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&#x27;s Law.
<ol>
	<li>The symmetry of the current is identified. For a non-symmetric current distribution, the magnetic field can be calculated using Biot Savart&#x27;s Law instead of Ampere&#x27;s Law.</li>
	<li>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.</li>
	<li>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.</li>
	<li>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.</li>
	<li>If the magnetic field is tangential to the path and the current enclosed is positive, then the magnetic field&#x27;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.</li>
	<li>Finally, the line integral of the magnetic field is equated with the current enclosed to get the magnetic field.</li>
</ol>

Ampere&#x27;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 ...

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Unit 1

Sources of Magnetic Fields

KEY TERMS AND CONCEPTS

choosing an amperian loop for magnetic fields

Ampere's law connects a closed loop path with the current enclosed by that loop. The line integral of the magnetic field around the path equals the vacuum permeability times the enclosed current. A right-hand rule helps set the sign of the current. If the fingers curl with the integration path, the thumb gives positive current. Current opposite the thumb direction is negative.

Solving magnetic field problems with Ampere's law starts by checking symmetry in the current distribution. If the current is not symmetric, Biot Savart's law is often used instead. When symmetry is present, a matching Amperian path is chosen. That path should be one where the magnetic field is constant or zero along the loop.

The best path is usually tangent to the magnetic field or perpendicular to it. When the magnetic field is constant and tangent to part of the loop, the line integral becomes the field strength times that path length. If the field is perpendicular to the path, or if the field magnitude is zero, that part of the integral is zero. The current enclosed by the loop is found by adding the individual currents that pass through the path.

The right-hand rule also gives the direction of the enclosed current. If the curl of the fingers follows the integration direction, the thumb points to positive current. If the magnetic field is tangent to the Amperian path and the enclosed current is positive, the field follows the integration direction. If the enclosed current is negative, the magnetic field points opposite the integration direction. The final step is to set the magnetic field integral equal to the enclosed current term and solve for the magnetic field.

Choosing an Amperian Loop for Magnetic Fields

choosing-an-amperian-loop-for-magnetic-fields

biot-savart law

Magnetic Field from a Current Element

The Biot-Savart law links a current-carrying wire to the magnetic field it creates. It gives both the magnitude and the direction of that field. The law is named for Jean-Baptiste Biot and Félix Savart, who studied how a straight wire carrying current interacts with a permanent magnet. A current-carrying wire produces a magnetic field around it. To describe the field from a tiny section of wire, we use an infinitesimal current element, dl. The direction of dl follows the direction of the...

biot-savart law: problem-solving

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Ampere's law connects a closed loop path with the current enclosed by that loop. The line integral of the magnetic field around the path equals the vacuum permeability times the enclosed current. A right-hand rule helps set the sign of the current. If the fingers curl with the integration path, the thumb gives positive current. Current opposite the thumb direction is negative. Solving magnetic field problems with Ampere's law starts by checking symmetry in the current distribution. If the...

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Toroid Magnetic Fields and Turn Count

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Magnetic Domains and Hysteresis

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key terms and concepts

Learn how to choose an Amperian loop, use the right-hand rule, and apply Ampere's law to solve magnetic field problems with symmetric current distributions.

Choosing an Amperian Loop for Magnetic Fields | JoVE K12

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New York State Science Learning Standards

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Choosing an Amperian Loop for Magnetic Fields

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