Consider a long wire where the current density is not uniform but instead increases as you approach the center of the wire, so that at a distance r from the center the current density is J(r)= I/ 2PiRr. Find the magnetic field strength both inside and outside of this wire.
I know I need to use biot-savart law, but im not sure how to do this with a non uniform density
Integrate the current from r = 0 to r = r, using the current density.
I(r) = Integral (r=0 to r) of J*2*pi*r dr
= I r/R where R is the radius of the wire
Then use Ampere's law that says the integral of H (=B/mu) around the loop equals the current flowing through the loop. The value of B will be uniform around that circular loop because of symmetry.
B(r) = mu*2*pi*(r/R)*(I/r)
= mu*2*pi*(I/R)
which is independent of r.
Outside the wire, use Ampere's law again:
B(r) = mu*2*pi(*I/r)
To find the magnetic field strength both inside and outside of the wire with non-uniform current density, you can indeed use the Biot-Savart law. The Biot-Savart law relates the magnetic field at a point P due to a small segment of current-carrying wire with a differential length element dl.
Let's break down the problem step by step:
Step 1: Determine the direction of the magnetic field:
Since the wire carries current, the magnetic field will form concentric circles around the wire's axis. Using the Right-Hand Rule, you can determine the direction of the magnetic field at a given point.
Step 2: Divide the wire into small current-carrying segments:
Divide the wire into small segments, each carrying a current I(r) = J(r) * 2πr * dl, where J(r) = I / (2πRr) is the given current density function.
Step 3: Calculate the magnetic field contribution from each segment:
Using the Biot-Savart law, the magnetic field dB at a point P due to a current-carrying segment of wire with length dl is given by:
dB = (μ₀ / 4π) * (I(r) * dl × r̂) / r²
Where μ₀ is the permeability of free space, I(r) is the current carried by the segment at radial distance r from the wire's center, dl is the differential length element, r̂ is the unit vector pointing from the segment to the point P, and r is the distance between the segment and point P.
Step 4: Integrate the contributions over the wire:
Integrate the magnetic field contributions from all the segments of the wire. The integral will depend on whether you are finding the field inside or outside the wire.
- Inside the wire: Integrate over the entire length of the wire, from -R to R.
- Outside the wire: Integrate over the length of the wire segment that contributes to the magnetic field at the point of interest.
Step 5: Apply symmetry and simplify the integral:
Given the current density function J(r) = I / (2πRr), note that the total current carried by a wire segment of length dl is constant along the wire. Simplify the integral using appropriate variable substitutions and symmetries.
After going through these steps, you should be able to determine the expressions for the magnetic field strength both inside and outside the wire, considering the non-uniform current density.