Question Video: Determining the Angle between a Current Carrying Coil and a Magnetic Field That Results in No Torque | Nagwa Question Video: Determining the Angle between a Current Carrying Coil and a Magnetic Field That Results in No Torque | Nagwa

# Question Video: Determining the Angle between a Current Carrying Coil and a Magnetic Field That Results in No Torque Physics • Third Year of Secondary School

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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?

02:14

### 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.

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