In this problem, we have a triangular loop in the x-y plane and a current-carrying wire is placed perpendicular to the plane. We need to find the emf induced in the loop when the current starts increasing at the rate of R.


To find the emf induced in the loop, we can use Faraday's law of electromagnetic induction which states that the emf induced in a closed loop is equal to the negative of the rate of change of magnetic flux through the loop.


To find the magnetic flux through the loop, we need to first find the magnetic field due to the current-carrying wire. Using the right-hand rule, we can see that the magnetic field lines will form circles around the wire. The magnitude of the magnetic field at a distance r from the wire can be given by the formula:

B = μ₀I/(2πr)

where μ₀ is the permeability of free space, I is the current in the wire, and r is the distance from the wire.

Now, let's consider each side of the triangular loop separately:

- Side 1: The magnetic field at the midpoint of this side will be perpendicular to the side and will have a magnitude of B = μ₀I/(2πd), where d is the distance from the wire to the midpoint of the side. The length of this side is l, so the magnetic flux through this side will be:

Φ₁ = B × l = μ₀Il/(2πd)

- Side 2: The magnetic field at the midpoint of this side will be at an angle of 60 degrees with respect to the side and will have a magnitude of B = μ₀I/(2πd), where d is the distance from the wire to the midpoint of the side. The length of this side is l, so the magnetic flux through this side will be:

Φ₂ = B × l × cos(60) = μ₀Il/(4πd)

- Side 3: The magnetic field at the midpoint of this side will be at an angle of -60 degrees with respect to the side and will have a magnitude of B = μ₀I/(2πd), where d is the distance from the wire to the midpoint of the side. The length of this side is l, so the magnetic flux through this side will be:

Φ₃ = B × l × cos(-60) = μ₀Il/(4πd)

The total magnetic flux through the loop will be the sum of the magnetic fluxes through each side:

Φ = Φ₁ + Φ₂ + Φ₃ = μ₀Il/(πd)


Now that we have found the magnetic flux through the loop, we can use Faraday's law to find the emf induced in the loop:

emf = -dΦ/dt = -μ₀Il/(πd) × R

where R is the rate of change of current.


Therefore, the emf induced in the loop is given by -μ₀IlR/(πd).