Using Ampere's circuital law, calculate the magnetic induction due a cylindrical current carryin

Using Ampere's circuital law, calculate the magnetic...

Key Concepts

Ampere's Circuital Law

Ampere's Circuital Law states that the line integral of the magnetic field around any closed loop is proportional to the net current enclosed by the loop. This law simplifies the calculation of magnetic fields in systems with high symmetry, such as infinite current-carrying conductors, by linking the geometry of the Amperian loop to the current through it.

Cylindrical Symmetry

Cylindrical symmetry is a key concept when dealing with infinite conductors, as it implies that the magnetic field has the same magnitude at any point equidistant from the axis of the cylinder. This symmetry allows one to choose a circular Amperian loop where the magnetic field is constant in magnitude and tangent to the loop, simplifying the integral in Ampere's Law.

Magnetic Field of an Infinite Straight Conductor

For an infinitely long current-carrying conductor, the magnetic field lines form concentric circles around the wire. The use of Ampere's Law combined with the cylindrical symmetry of the system leads to an expression where the magnetic field magnitude decreases inversely with the distance from the conductor. The direction of the magnetic field is determined by the right-hand rule.

Current Enclosure

In applying Ampere's Law, determining the net current enclosed by the chosen Amperian loop is essential. Only the current within the loop contributes to the magnetic field calculation, making it critical to correctly identify and account for the geometry of the current distribution in the conductor.

Transcript

00:01 We're going to be finding the magnetic field of a wire at a position are less than the radius of the wire.

00:12 So we're given the current density inside the wire and to recognize that current density is equal to the current per unit area that is perpendicular to the current flow.

00:31 So here we have a variable current density.

00:35 It is the dentist in the center of the wire where the radius is zero, and it just linearly goes down from that point onwards.

00:46 But anyway, the way to approach us is to use amper's law, which says that if you have a loop, a path that goes all the way around in a circuit, that the magnetic field projected along to that path is equal to mu not times the current enclosed by that path.

01:12 And because of the cylindrical symmetry, the path that we want to draw is a circular path inside the wire, because we want to find the magnetic field inside the wire.

01:27 And so what we'll have to find is the current enclosed by that loop.

01:34 Which is going to be proportional to the integral of the current density times the area of that loop.

01:46 So it's kind of counterintuitive, but we actually don't have to do any integration on the left -hand side.

01:54 We're assuming that b is constant along that circular path, and going all the way around, it has a length of 2 pi r, where that little r, let me see if i can draw that in red or something, is the radius out from the center to the imaginary loop.

02:17 So the wire has a real radius that ends the wire, but here we're going out to the loop because we want to find the magnetic field at an arbitrary point inside.

02:31 For the current enclosed, we are going to have to take our current density and multiply it by the area of that blue area.

02:49 So it's a circle, and we can write down the area is rdr times 2 pi, 2 pi rdr.

03:02 So we're kind of making rings inside of that blue area, rings that have a circumference of 2xr, and i really should call it r -prime, and they have width dr prime, and we are going to integrate from 0 to r.

03:31 And that should get everything inside that blue area.

03:34 So yeah, it's kind of counterintuitive that the integral usually winds up on the right -hand side of ampeer's law...

Please give Ace some feedback