Two concentric circular loops of wire lie on a tabletop, one inside the other. The inner wire has a diameter of 18.0 cm and carries a clockwise current of 14.0 A, as viewed from above, and the outer wire has a diameter of 40.0 cm. What must be the magnitude and direction (as viewed from above) of the current in the outer wire so that the net magnetic field due to this combination of wires is zero at the common center of the wires?

The current flows clockwise, so curl all of your fingers except your thumb clockwise.
Your thumb will point into the page, therefore the magnetic field points into the page.

We want the magnetic field due to the outer wire to cancel out the field due to the inner wire. Use the right-hand rule again:

The field from the outer wire should point out of the page to counter the field from the inner wire. Point your thumb out of the page.
Curl your other fingers. They should curl counterclockwise, therefore the current in the outer wire should flow counterclockwise.

The current in the outer wire should flow counterclockwise.

The magnetic field strength at the center of a circular current-carrying loop is given by:

B = 0.5μ₀I/R

B = magnetic field strength, μ₀ = magnetic constant, I = current, R = radius

μ₀ = 4π×10⁻⁷H/m

Given values for the inner wire:

I = 14.0A, R = (18.0cm)/2 = 9.00×10⁻²m

Given values for the outer wire:

R = (40.0cm)/2 = 20.0×10⁻²m

Since the net magnetic field at the center should be zero, calculate the magnetic field due to the inner and outer wires, set them equal to each other, and solve for I, the current in the outer wire:

0.5(4π×10⁻⁷)(14.0)/(9.00×10⁻²) = 0.5(4π×10⁻⁷)I/(20.0×10⁻²)

14.0/(9.00×10⁻²) = I/(20.0×10⁻²)

I = 31.1A