Introductory Physics Volume Two

106 Sources of Magnetic Field 5.4
Example
Finite Wire
A straight wire of length a carries a current I. What is the magnetic
field caused by the current at a point (x, y) measured from the center
of the wire?
What is B here?
x
y
The wire is a straight line, which can be parametrized:
⃗r s = (at)î,
where t varies from − 1 2 to + 1 2
. So we have:
d⃗r s
dt = aî
We want to find the magnetic field at the position
⃗r f = xî + yĵ,
Collect the quantities needed for applying the Biot-Savart law:
⃗r f − ⃗r s = (x − at)î + yĵ
Using the Biot-Savart law:
| ⃗r f − ⃗r s | = √ (x − at) 2 + y 2
d⃗r s
dt × (⃗r f − ⃗r s ) = (aî) × ((x − at)î + yĵ)
⃗B(x, y) = µ 0I
4π
= µ 0I
4π
= ayˆk
∫
1
∫2
− 1 2
= − µ 0Iy
4π
d⃗rs
dt × [⃗r f − ⃗r s ]
| ⃗r f − ⃗r s | 3 dt
ayˆk
dt
((x − at) 2 + y 2 ) 3 2
x−
∫
a 2
x+ a 2
du
(u 2 + y 2 ) 3 2
where I have substituted u = x − at into the last expression. Looking
ˆk