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 c…
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.
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Q1: How does Ampere's Law apply to finding the magnetic field inside a thick conductor?
For a thick conductor with uniform current, use an Amperian loop with radius smaller than the conductor radius. The line integral of the magnetic field equals the product of magnetic field and loop circumference. The enclosed current equals current density times the enclosed area. Applying Ampere's Law yields a magnetic field that increases linearly from the conductor's center to its surface.
Q2: What is the relationship between magnetic field and distance outside a current-carrying conductor?
Outside a thick conductor, use an Amperian loop with radius greater than the conductor radius. The enclosed current equals the total current flowing through the conductor. Applying Ampere's Law shows the magnetic field is inversely proportional to the loop radius, meaning field strength decreases as distance from the conductor increases.
Q3: Why is cylindrical symmetry important when applying Ampere's Law to a straight conductor?
Cylindrical symmetry ensures that magnetic field lines form concentric circles around the conductor axis, making them either constant or zero along a symmetric Amperian path. This symmetry allows you to choose an integration path where the magnetic field is tangential or perpendicular, simplifying the line integral calculation and enabling direct application of Ampere's Law.
Q4: How do you determine the direction of the magnetic field using the right-hand rule in Ampere's Law?
Curl your right hand's fingers along the integration path direction. Your thumb points in the positive current direction. If enclosed current is positive and the magnetic field is tangential to the path, the field direction follows the integration direction. If enclosed current is negative, the magnetic field direction is opposite to the integration direction.
Q5: When should you use Biot-Savart Law instead of Ampere's Law for magnetic field calculations?
Use Biot-Savart Law when the current distribution lacks symmetry. Ampere's Law requires identifying symmetric current distributions and choosing symmetric integration paths where the magnetic field is constant or zero. For non-symmetric current distributions, Biot-Savart Law provides a more general approach to calculating magnetic fields without relying on symmetry.
Q6: What happens to the magnetic field magnitude at the surface of a thick conductor?
At the conductor's surface, the magnetic field reaches its maximum value. Inside the conductor, the field increases linearly from the center to this maximum at the surface. Outside the conductor, the field magnitude drops inversely with distance. This transition occurs because the enclosed current changes from increasing linearly inside to remaining constant outside.
Q7: How does the line integral simplify when the magnetic field is tangential to an Amperian path?
When the magnetic field is tangential to the Amperian path and constant along that portion, the line integral reduces to the product of the constant magnetic field magnitude and the path length for that section. For regions where the magnetic field is perpendicular to the path or has zero magnitude, the line integral contribution is zero, allowing you to focus only on tangential segments.