Magnetic field due to a current carrying loop

[ajay.05]

Homework Statement

The actual question was to somehow derive the expression for a magnetic field at the center of a current carrying loop.

Homework Equations

The Biot-Savart's law equation
dB = (mu/4π)*(i dl x r)/r^3
(I'm not actually good at typing in the equations, so please just forgive me)

The Attempt at a Solution

[/B]
Found an expression for the magnetic field due to a differential element at the centre and integrated it along the loop, and I got the expression.
Now, exactly coming to my point, is this derivation possible using Ampere's circuital law? If yes, how? If no, then what are the other situations except the cases of solenoid and toroid, where it is applicable? Please be brief and thanks in advance...

 

[kuruman]

You cannot use Ampere's law for the circular loop because the symmetry is not high enough. You can use Ampere's law to find the the B field due to (a) a very long wire and its variants (long rod, strip of finite width) and (b) infinite sheet of current. I cannot think of anything else right now.

 

[twist.1995]

Ampere's Law for finding magnetic fields in primarily useful when you deal with high-symmetry systems. Examples of these systems are: the magnetic field perpendicular to the direction of the current in a straight wire, the magnetic field of the thick cylindrical wire, having uniform charge density, the field of a solenoid or a toroid, the magnetic field of a current conducting plane (thin plate) (contribution of infinitely many infinitesimal wires at some point). The important point about applying Ampere's law is that the magnetic field has to be circulating around some current element. The direction of \vec ds is in the same direction with \vec dB, or, in other words, these vectors are parallel. The integration involves the total current through the area, and you can use the right hand rule to find how the magnetic field circulates (thumb indicates the direction of the current, and the curl of your other fingers will give the direction of the magnetic field). On the other hand, if you consider a solenoid, the four fingers give the direction of the current through the windings, and the thumb indicates the direction of the magnetic field (the same principle applies to the toroid). The Biot-Savart law is more "Universal", and it applies to asymmetrical systems, when Ampere's Law fails to work. Keep in mind that in the case of Biot-Savart law, the vector \vec dB is perpendicular to both \hat {\mathbf r} and \vec ds. You start evaluating the Biot-Savart law by taking the cross product of \hat {\mathbf r} and \vec ds. When you have bended single wires, straight wires of the finite length, a circular current conducting loop, you will generally apply Biot-Savart law.

Hint: the vector \hat {\mathbf r} is perpendicular to \vec ds when it is a circular loop. Also, keep in mind that the cross product gives another vector perpendicular to \hat {\mathbf r} and \vec ds. The small element \vec ds can be expressed as R\vec d\theta. So, your integral will involve only the angle of revolution \theta.

 

[ajay.05]

Ampere's Law for finding magnetic fields in primarily useful when you deal with high-symmetry systems. Examples of these systems are: the magnetic field perpendicular to the direction of the current in a straight wire, the magnetic field of the thick cylindrical wire, having uniform charge density, the field of a solenoid or a toroid, the magnetic field of a current conducting plane (thin plate) (contribution of infinitely many infinitesimal wires at some point). The important point about applying Ampere's law is that the magnetic field has to be circulating around some current element. The direction of \vec ds is in the same direction with \vec dB, or, in other words, these vectors are parallel. The integration involves the total current through the area, and you can use the right hand rule to find how the magnetic field circulates (thumb indicates the direction of the current, and the curl of your other fingers will give the direction of the magnetic field). On the other hand, if you consider a solenoid, the four fingers give the direction of the current through the windings, and the thumb indicates the direction of the magnetic field (the same principle applies to the toroid). The Biot-Savart law is more "Universal", and it applies to asymmetrical systems, when Ampere's Law fails to work. Keep in mind that in the case of Biot-Savart law, the vector \vec dB is perpendicular to both \hat {\mathbf r} and \vec ds. You start evaluating the Biot-Savart law by taking the cross product of \hat {\mathbf r} and \vec ds. When you have bended single wires, straight wires of the finite length, a circular current conducting loop, you will generally apply Biot-Savart law.

Hint: the vector \hat {\mathbf r} is perpendicular to \vec ds when it is a circular loop. Also, keep in mind that the cross product gives another vector perpendicular to \hat {\mathbf r} and \vec ds. The small element \vec ds can be expressed as R\vec d\theta. So, your integral will involve only the angle of revolution \theta.

Wow! Thanks, but what does that 'high symmetry' here signify?

 

[twist.1995]

Wow! Thanks, but what does that 'high symmetry' here signify?

The current flows in the cylindrical surface, long rectangular plane, straight wire, or any other geometrical shapes that will give you circular loops of the magnetic field. Normally, the current is normal to a cross-sectional area at any time and it passes through the loops around which the magnetic field is created.