1
Magnetic Field of a Current Loop
P
a
x
dB
x
θ
A circular ring of radius a carries a current I as shown. Calculate the magnetic field at a point P along the axis of the ring at a distance x from its center.
dl
90-θ
I
y
dBy
dBx
θ
90-θ
z
Complicated diagram! You are supposed to visualize the ring lying in the yz plane.
dl is in the yz plane. r is in the xy plane and is perpendicular to dl. Thus
Also, dB must lie in the xy plane (perpendicular to dl) and is also perpendicular to r.
2
P
a
x
dB
x
θ
dl
90-θ
I
y
dBy
dBx
θ
90-θ
z
By symmetry, By will be 0. Do you see why?
3
P
x
dBz
x
dl
I
y
dBy
dBx
z
When dl is not centered at z=0, there will be a z-component to the magnetic field, but by symmetry Bz will still be zero.
4
I, x, and a are constant as you integrate around the ring!
P
a
x
dB
x
θ
dl
90-θ
I
y
dBy
dBx
θ
90-θ
z
5
At the center of the ring, x=0.
P
a
x
dB
x
θ
dl
90-θ
I
y
dBy
dBx
θ
90-θ
z
For N tightly packed concentric rings (a coil)…
6
Magnetic Field at the center of a Current Loop
A circular ring of radius a lies in the xy plane and carries a current I as shown. Calculate the magnetic field at the center of the loop.
a
z
dl
I
x
y
The direction of the magnetic field will be different if the plane of the loop is not in the xy plane.
I’ll explain the “funny” orientation of the axes in lecture.