DC Motor Principles: Torque on a Current Loop

Hello!

In our last lesson, we explored how mechanical work is converted into electrical energy in a generator. We found that a crucial part of this process is the counter-torque experienced by the current-carrying coil. To keep the generator spinning, a turbine must do mechanical work to overcome this torque.

Today, we're going to flip that concept on its head. What if producing this torque isn't a cost, but the entire goal? This simple change in perspective takes us from a generator to an electric motor. This lesson will focus on the learning outcome: Explain the operating principle of a DC motor in terms of the torque on a current-carrying loop in a magnetic field.

We will break down how supplying an electric current to a coil placed in a magnetic field creates a rotational force, and how this is harnessed to produce continuous motion.

From Force to Torque

The operating principle of a motor starts with a concept we've already encountered: the Lorentz force. A wire carrying a current $I$ through a magnetic field $\vec{B}$ experiences a force. Your background in vector calculus might remind you this is a cross product, $\vec{F}=I\vec{L}\times \vec{B}$, where $\vec{L}$ is the length vector of the wire. This means the force is always perpendicular to both the current and the magnetic field.

Now, imagine a simple rectangular loop of wire placed in a uniform magnetic field, as shown below.

Let's analyze the forces on each segment of the loop:

• Sides parallel to the field: The current flows parallel or anti-parallel to the magnetic field lines. The angle $\theta$ between $\vec{L}$ and $\vec{B}$ is 0° or 180°. Since $\sin (0^{\circ })=\sin (180^{\circ })=0$, these segments experience no magnetic force.
• Sides perpendicular to the field: These two sides have current flowing in opposite directions. Using the right-hand rule, you'll find they experience forces that are also in opposite directions—one side is pushed up, and the other is pushed down.

Because these two equal and opposite forces are applied at a distance from the central axis of rotation, they form a couple, creating a net torque that causes the coil to rotate.

The following video provides an excellent visual demonstration of this principle.

Quantifying the Torque

To understand the motor's behavior, we need to quantify this torque. The magnitude of the torque ($\tau$) depends on several factors: the strength of the magnetic field ($B$), the current ($I$), the area of the loop ($A$), and the number of turns of wire in the coil ($N$).

Crucially, the torque also depends on the orientation of the coil relative to the magnetic field. The torque is maximum when the forces are applied at the greatest possible lever arm, which occurs when the plane of the coil is parallel to the magnetic field. The torque becomes zero when the coil is perpendicular to the field, as the forces on the sides pull directly against the axis of rotation, creating no turning effect.

This relationship is captured by the following equation:
$\tau =NIAB\sin (\theta )$
Here, $\theta$ is the angle between the magnetic field vector $\vec{B}$ and the normal vector of the coil's area (a vector pointing perpendicular to the face of the coil).

The following segment of the same video derives this equation and introduces the useful concept of the magnetic dipole moment, $\mu =NIA$, which simplifies the torque equation to $\tau =\mu B\sin (\theta )$.

For a more formal treatment using vector notation, which you may find insightful, you can refer to Section 2 of the LibreTexts article "Force and Torque on a Current Loop". It derives the torque as a cross product, $\vec{\tau }=\vec{\mu }\times \vec{B}$, which elegantly captures both the magnitude and direction.

From Oscillation to Continuous Rotation: The Commutator

We now have a problem. As the coil rotates past the vertical position ($\theta =0^{\circ }$), the direction of the torque reverses, pushing it back the way it came. Without any changes, the coil would simply oscillate around the vertical position.

To build a functional motor, we need to ensure the torque always pushes the coil in the same rotational direction. The ingenious solution for a DC motor is the split-ring commutator.

The commutator is a ring split into two halves, with each half connected to one end of the coil. Fixed contacts, called brushes, press against the rotating commutator, supplying power from the DC source.

Here’s the key action:

1. As the coil rotates, the torque pushes it in one direction (e.g., clockwise).
2. Just as the coil passes the vertical position (where torque would naturally reverse), each brush slides from one half of the commutator to the other.
3. This action reverses the direction of the current flowing through the coil.
4. By reversing the current just as the forces are about to reverse the torque, the torque continues to push the coil in the original direction (clockwise).

Instead of a reversing torque $\tau =NIAB\sin (\theta )$ (which would be a standard sine wave), the commutator effectively gives us a torque of $\tau =NIAB|\sin (\theta )|$. This results in a pulsating, but always positive, torque, driving continuous rotation. The graph of torque versus rotation angle would look like a series of rectified sine humps, never going negative.

Conclusion

We have successfully deconstructed the DC motor, seeing how it cleverly engineers the Lorentz force into a continuous rotational output.

Key Takeaways:

• The fundamental principle of an electric motor is that a current-carrying loop in a magnetic field experiences a torque.
• This torque arises from the magnetic forces acting on the sides of the loop, which form a couple that causes rotation.
• The magnitude of the torque is described by the equation $\tau =NIAB\sin (\theta )$, which is at a maximum when the coil's plane is parallel to the field and zero when perpendicular.
• To achieve continuous rotation, a DC motor uses a split-ring commutator and brushes to reverse the direction of the current in the coil every half turn, ensuring the torque always acts in the same rotational direction.

Preview of the Next Lesson:

We've established the principle and the core equation for motor torque. In the next lesson, we will focus on the next learning outcome: Calculate the torque on a motor's coil from current, magnetic field strength, and geometry. We will apply the torque equation to practical examples, giving you the tools to analyze the performance of a motor quantitatively.