Magnetic Field: Inside & Outside Current Carrying Wire

[dontwannatellyou]

Hi
My question is about magnetic field I mean I know where their formulas are derived from amperes law but I didnt really understand why and how there's is a difference in magnetic fields outside and inside a crrent carrying wire
the field inside a current carrying wire is B = μoIr^2/2πR^2
the field outside a current carrying wire is B =μoI/2πR

 

[ZapperZ]

Hi
My question is about magnetic field I mean I know where their formulas are derived from amperes law but I didnt really understand why and how there's is a difference in magnetic fields outside and inside a crrent carrying wire
the field inside a current carrying wire is B = μoIr^2/2πR^2
the field outside a current carrying wire is B =μoI/2πR

Usually, when someone is at the stage of doing magnetic field, that person should have already done problems in electric field, especially in the use of Gauss's Law.

I will assume that you have, because I'm going to ask you something similar to your problem. If you have a sphere with a uniform charge, haven't you seen the difference between the expression for the E-field inside the sphere and outside the sphere?

Zz.

 

[Nolan]

The difference in the two answers you provided is the result of a fundamental assumption about how current flows through a wire. The problem you're dealing with here assumes that the current is evenly distributed over the wire's cross section. When you use Ampere's Law inside of the wire, you are looking to find how much of the total current (I) is enclosed by your circular loop which has a radius (r), which is less than the radius of the wire (R). Any current outside of this loop (which has a radius less than that of the wire) is not "enclosed" and does not contribute.

To do this, you take the ratio between the area enclosed by the loop and the wire's cross sectional area and multiply it by the total current. I(enclosed) = I(total) x [(pi*r^2)/(pi*R^2)]. This expression for I(enclosed) is what you put into Ampere's Law. If the current is not constant and is some function of r, you would have to integrate the given expression over the cross sectional area (from r = 0, to some value r=r).

REMEMBER, you can only use the ratio of the areas to find the enclosed current because the current has no r dependence (s if you are in cylindrical coordinates).

Also, the expression you gave for the magnetic field inside the wire is almost correct... When you use the expression for I(enclosed) we found earlier, you will find that a small r cancels from your expression of Ampere's Law: B (2*pi*r) = [I(total) x (pi*r^2)/(pi*R^2)] (μ0). Here the pi's cancel and the r^2 is reduced to a single r in the numerator.