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 equal to the permeability of free space multiplied by the net current enclosed by that loop. This law is crucial for calculating magnetic fields in systems with high symmetry, serving as a fundamental tool in classical electromagnetism.

Cylindrical Symmetry

Cylindrical symmetry implies that the physical properties of the system are invariant under rotations about the conductor's axis. This symmetry allows the magnetic field to be constant along circular paths centered on the axis, significantly simplifying the computation of the field using Ampere's Law.

Current Density

Current density is defined as the current per unit area and is an essential concept when dealing with uniformly distributed currents. In the context of a cylindrical conductor, it helps determine the fraction of the total current enclosed by an Amperian loop located inside the conductor, thereby facilitating the calculation of the internal magnetic field.

Magnetic Field Inside a Conductor

The magnetic field inside a current-carrying conductor depends on the portion of the total current that is enclosed by the chosen Amperian loop. For a uniformly distributed current, the field at a point inside the conductor is calculated by considering the ratio of the area inside the loop to the total cross-sectional area of the conductor, linking directly back to the integral form of Ampere's Law.

Transcript

0:00 Hi there.

00:01 So for this problem, we are told that a wire of radius r carries a current i, and the current density is given by the following equation that is equal to the initial current density times 1 minus the radius over the radius of the wire, where r is measured from the center of the wire, and j sub 0.

00:32 Is a constant.

00:34 So we need to use unpers law to find the magnetic field inside the wire at a distance r that is less than the radius of the wire.

00:48 So with that said, let me just draw the situation in here.

00:55 So we have this wire, this cross -ceptional of the wire, and this is with a radius capital r and we are going to have an umpare an imperial loop inside this wire with a radius art and we know that in here the current density depends on the radios so what we are going to do is to apply um per's law so from that we are going to obtain that the um the close integral the close integral between the product between the magnetic field and the ds, which corresponds to a segment of this umpary loop, is going to be equal to, well, we can take out the magnetic field because it is constant, and we will have the integral of the ds.

02:05 If we integrate all of the ds, we are going to take the perimeter of the, and umpary loop.

02:12 So we are going to obtain the magnitude of the magnetic field times 2 times pi times the radius are.

02:20 And this is going to be equal by umpers law equals to new sub zero, which is a constant times the current and close by the um the imperian loop.

02:34 So the close current is going to be the, integral of the current density times the differential in area.

02:51 So we can write this as the integral from 0 to 2 pi because we are integrated all of that angle from 0 to pi and from 0 to the radius art.

03:05 And in here we are going to have the current density.

03:10 We're going to call it prime, r prim to differentiate.

03:13 And this r prime, the r prime, and the theta.

03:20 So we know that the current density is a function of the position r, only about that.

03:33 So the above integral, this integral is going to be, we can do the integral from 0 to 2 pi because nothing depends in here on tita...

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