Magnetic Field from a Current Segment
The notes outline how to find the magnetic field from a very long wire of current. Now, what is the magnetic field from a single segment? Suppose we have the configuration shown below. Your observation point is at the origin, and the segment of current I runs in a straight line from ⟨−L,0,0⟩ to ⟨0,−L,0⟩.
Facts
- The current in the segment is I.
- The observation point is at the origin.
- The segment stretches from from ⟨−L,0,0⟩ to ⟨0,−L,0⟩.
Lacking
- →B
Approximations & Assumptions
- The current is steady, and the wire segment is uniform.
Representations
- We represent the Biot-Savart Law for magnetic field from a current as
→B=∫μ04πI⋅d→l×→rr3
- We represent the situation with diagram given above.
Solution
Below, we show a diagram with a lot of pieces of the Biot-Savart Law unpacked. We show an example d→l, and a separation vector →r. Notice that d→l is directed along the segment, in the same direction as the current. The separation vector →r points as always from source to observation.
For now, we write d→l=⟨dx,dy,0⟩
If take the derivative of the line equation y=−x−L then we can figure out how dx relates dy. (Use geometry logic instead) This would give us dy=−dx. We can now plug in to express d→l and →r in terms of x and dx:
d→l=⟨dx,−dx,0⟩