Consider a thin, straight wire carrying a constant current I and placed along the y-axis as shown in Figure. Determine the magnitude and direction of the magnetic field at point P on x axis due to this current if the direction of current is in y axis.
From the Biot–Savart law, we expect that the magnitude of the field is proportional to the current in the wire and decreases as the distance a from the wire to point P increases.
We are asked to find the magnetic field due to a simple current distribution, so this example is a typical problem for which the Biot–Savart law is appropriate. We must find the field contribution from a small element of current and then integrate over the current distribution.
Let’s start by considering a length element located a distance r from P. The direction of the magnetic field at point P due to S the current in this element is out of the page because is out of S the page. In fact, because all the current elements lie in the plane of the page, they all produce a magnetic field directed out of the page at point P. Therefore, the direction of the magnetic field at point P is out of the page and we need only find the magnitude of the field. We place the origin at O and let point P be along the positive y axis, with being a unit vector pointing out of the page.
Evaluate the cross product in the Biot–Savart law:
From the geometry in Figure, express r in terms of u:
Notice that tan from the right triangle in Figure (the negative sign is necessary because is located at a negative value of x) and solve for x:
Find the differential dx:
Substitute equations into the magnitude of the field:
Integrate equation over all length elements on the wire, where the subtending angles range from θ1 to θ2 as defined in Figure:
We can use this result to find the magnetic field of any straight current- carrying wire if we know the geometry and hence the angles θ1 and θ2. Consider the special case of an infinitely long, straight wire. If the wire in b becomes infinitely long, we see that for length elements ranging between positions . Because
The magnitude of the magnetic field is proportional to the current and decreases with increasing distance from the wire, as expected.
Comments