### Video Transcript

For a current-carrying coil in a
uniform magnetic field, what is the angle between the coil’s plane and the direction
of the magnetic field that results in no torque acting on the coil?

To answer this question, it’s
important to distinguish between applied force that could act to produce rotation of
the coil and forces that act on wires of the coil due to current in the wires and
the presence of the uniform magnetic field, which is what this question
concerns. It’ll be helpful to draw a diagram
to help illustrate this.

We should recall that for a
current-carrying rectangular loop of wire in a magnetic field, the torque 𝜏 is
equal to 𝐵𝐼𝐴𝑁 sin 𝜃, where 𝐵 is the magnitude of the magnetic field, 𝐼 is the
current in the coil, 𝐴 is the surface area of the coil, 𝑁 is the number of turns
in the coil, and 𝜃 is the angle between the normal to the surface area of the coil
and the direction of the magnetic field.

Now, it’s very important to
understand that the direction of the plane of the coil is not the direction of the
normal to the surface area of the coil. The plane of the coil is
perpendicular to the normal to the surface area of the coil. We see then that the angle 𝜃 is
not the angle that the question is concerned with. Rather, the question concerns the
angle between the plane of the coil and the magnetic field, and this is the angle
𝜑.

The question requires that no
torque acts on the coil, so the torque, 𝜏, must be equal to zero. That is, 𝐵𝐼𝐴𝑁 sin 𝜃 equals
zero. That equality is zero if and only
if the sin of 𝜃 equals zero, because all the other quantities in the equation are
nonzero. Then, we can deduce that 𝜃 equals
zero degrees or 180 degrees.

Because of this, we know that the
normal to the area 𝐴 must be parallel to the direction of the magnetic field
𝐵. As we stated earlier, by
definition, the normal to the area 𝐴 is perpendicular to the plane that contains
the area 𝐴. Thus, we know that the angle
between the plane containing 𝐴 and the normal to the surface is 90 degrees. And therefore, the angle 𝜑 between
the plane containing the area 𝐴 and the magnetic field is perpendicular when the
torque is equal to zero, that is, when 𝜑 equals 90 degrees. Therefore, the correct answer is 90
degrees.