# Ampere's Law For Static Magnetic Field

#### BlackMelon

Joined Mar 19, 2015
168
Hi there!

Please refer to the picture below. I would like to understand the equation Curl(H) = J, where H is the magnetic field intensity and J is the current density. So, I inspect a simple problem.
There is a wire carrying current I in the z-axis direction. a_r, a_phi, and a_z are the unit vectors in the directions of the radius, the tangential line, and z-axis, respectively.

So, from H = I/(2*pi*r)a_phi. I take the curl of this vector (in cylindrical coordinate) and got 0. How does this relate to the current density?

Best
BlackMelon

#### MrAl

Joined Jun 17, 2014
11,268
Hi there!

Please refer to the picture below. I would like to understand the equation Curl(H) = J, where H is the magnetic field intensity and J is the current density. So, I inspect a simple problem.
There is a wire carrying current I in the z-axis direction. a_r, a_phi, and a_z are the unit vectors in the directions of the radius, the tangential line, and z-axis, respectively.

So, from H = I/(2*pi*r)a_phi. I take the curl of this vector (in cylindrical coordinate) and got 0. How does this relate to the current density?

Best
BlackMelon

View attachment 303731
Hello,

Try writing out your full solution method in detail. That is, what you multiplied and what you subtracted or added. You should get three different parts with two terms in each part. Once you have it written out, go over it and see if you made a mistake in calculating the curl.
For example in this form:
(a-b)+(c-d)+(e-f)
however they may not be all added some or all may be subtracted.
Each variable a through f will also be a multiplication such as:
a=A*B
b=C*D
c=E*F
etc.
It is obvious that some of those multiplications will be zero, but don't do it in your head, do it on paper (or text or graphic image like you did above).
Once you write it all out we can go over it.

#### BlackMelon

Joined Mar 19, 2015
168
Hello,

Try writing out your full solution method in detail. That is, what you multiplied and what you subtracted or added. You should get three different parts with two terms in each part. Once you have it written out, go over it and see if you made a mistake in calculating the curl.
For example in this form:
(a-b)+(c-d)+(e-f)
however they may not be all added some or all may be subtracted.
Each variable a through f will also be a multiplication such as:
a=A*B
b=C*D
c=E*F
etc.
It is obvious that some of those multiplications will be zero, but don't do it in your head, do it on paper (or text or graphic image like you did above).
Once you write it all out we can go over it.

I got the answer for my misunderstanding. The curl is defined point by point. For example, the result of curl of H(x,y,z) is a vector at (x,y,z).
H = I/(2*pi*r)a_phi is the magnetic field outside the wire. From Curl(H) = J, since there is no current flowing outside the wire, it is zero.

By the way, see the picture below. I have analyzed the inside of the wire H = I*r/(2*pi*R^2) a_phi.
where r is the radius from the center of the wire to the point of interest. R is the radius of the wire. And got the correct answer:
J = curl (H) = I/(pi*R^2) a_z