The Magic of Reciprocity
Imagine trying to calculate the magnetic flux passing through an infinitely long straight wire due to a triangular loop. It sounds like a mathematical nightmare! But physics offers us a beautiful shortcut: the Reciprocity Theorem of Mutual Inductance. This theorem states that the mutual inductance between two circuits is symmetric, meaning M12=M21. Instead of tackling the complex field of the triangle, we can simply assume a steady current I in the straight wire and calculate the flux it produces through the triangular loop.
Slicing the Triangle
Let's set up our geometry. The straight wire lies along the x-axis. The right-angled isosceles triangle has its 90∘ vertex at the origin, extending downwards. The magnetic field produced by the wire at a distance y is given by Ampere's Law: B=2πyμ0I.
To find the total flux, we slice the triangle into horizontal elemental strips of thickness dy. Because the vertex angle is 90∘, the boundaries of the triangle are the lines y=x and y=−x. Thus, at a depth y, the width of the strip is exactly 2y. The area of this strip is dA=2ydy.
The Master Equation
Now, we calculate the small magnetic flux dϕ passing through this strip:
dϕ=BdA=(2πyμ0I)(2ydy)=πμ0Idy
Notice how beautifully the y terms cancel out! Integrating this from y=0 to the height of the triangle y=h, we get the total flux:
The mutual inductance is simply M=Iϕ=πμ0h.
Given that the current in the loop is changing at a rate of dtdi=10 A/s and the height h=0.1 m, the induced EMF in the wire is:
e=Mdtdi=(πμ0(0.1))(10)=πμ0 V
This confirms that the magnitude of the induced EMF is indeed πμ0 volts.
Lenz's Law and Repulsion
What about the force between them? Enter Lenz's Law. The induced current always acts to oppose the change in magnetic flux. Since the current in the loop is increasing, the induced current in the wire will flow in a direction that attempts to push the loop away, reducing the mutual flux linkage. This results in a repulsive force between the wire and the loop.
The Symmetry of Rotation
Finally, consider rotating the loop around the wire. The magnetic field of a straight wire possesses perfect cylindrical symmetry. As the loop rotates at a constant radius, the magnetic field lines it cuts do not change, meaning the magnetic flux remains perfectly constant. Since the rate of change of flux is zero (dtdϕ=0), no additional EMF is induced during this rotation.
