Magnetic Field inside and external to a wire

[Gear300]

I again bring a question: If a wire of radius R carries a current density J = br (r is radius and b is a constant), in which J = I/A (A is area)...then derive an expression for the magnetic field at r1 (r1 being a radial distance less than R) and at r2 (r2 being a radial distance greater than R).

My answer at r1 is B = (u*b*r1^2)/2 and at r2 is B = (u*b*R^3)/(2*r2), in which u is the permeability of free space.
The actual answer at r1 is B = (u*b*r1^2)/3 and for r2 is B = (u*b*R^3)/(3*r2)...which seem to match my answers...just instead of halving each one...its divided by 3...how did they get that?

 

[Gear300]

Actually...nevermind...I found out why...I apparently had to integrate for increments of I.

 

[Moetasim]

There are a few factors that could have led to the discrepancy between your answer and the actual answer for the magnetic field inside and outside a wire.

Firstly, it is important to note that the expression for the magnetic field inside and outside a wire is derived using the Biot-Savart Law, which is based on an idealized scenario of an infinitely long straight wire. In reality, wires have finite lengths and may also have bends or curves, which can affect the magnetic field distribution.

Additionally, the current density in a wire is not constant throughout its cross-sectional area. The current density is typically highest at the center of the wire and decreases towards the edges. This non-uniform current distribution can also affect the magnetic field strength at different radial distances.

Furthermore, the expression for the magnetic field at a particular radial distance is also dependent on the chosen reference point. The Biot-Savart Law assumes that the reference point is located at infinity, which may not always be the case in practical scenarios.

It is possible that your calculations may have considered a different reference point or a slightly different scenario, leading to a slight discrepancy in the results. However, the fact that your answers are in the same ballpark as the actual answers suggests that your understanding of the concept is correct, and any discrepancies can be attributed to minor differences in assumptions and calculations.