Introductory Physics Volume Two

110 Sources of Magnetic Field 5.4
since d ⃗ l and the radius vector to the point are perpendicular. The
length of dl can be approximated by the length of the arc subtended
by the angle ∆θ, so
|dB| ⃗ = Iµ 0 (adθ)(a)
4π a 3 ,
= Iµ 0 dθ
4π a
To find the magnetic field due to the entire wire, integrate:
B = Iµ 0
4π
∫ π
0
dθ
a = Iµ 0
4a
Example
Two sections of straight wire carrying electric current lie parallel to
each other, a distance b apart:
I 2
b
a
I 1
The leftmost ends of the wires are aligned. What is the force that
current I 1 exerts on current I 2 ? Draw a coordinate system:
y
dx'
x'
I 2
b
I 1
x
x=-
a
2
The z-axis points out of the page. From a previous example, we know
that the magnetic field due to I 1 in the xy-plane is
⎛
⎞
⃗B 2 (x, y) = µ 0I 1 x +
⎝
a 2
x − a 2
√
4πy (x )
− √ ⎠ ˆk
+
a 2
2 + y
2 (x −
a
2 )2 + y 2