Lenz's Law

Lenz's Law Definition

In order to understand how Lenz's law plays a part in electromagnetic induction, we must first define the phenomenon of induction.

We can also define EMF as the following.

Electromagnetic induction is a key phenomenon in technologies we use in our everyday lives; as we mentioned in the introduction, induction cookers generate heat in cooking pans by inducing an electromotive force in the ferrous metal, rather than using fire. This electromotive force generates a current running through the pan which then, due to the resistance of the material, creates heat, allowing you to cook your breakfast!

What Lenz's law does is establish the direction of the induced electromotive force in the conductor.

So what do we mean by "opposes the motion causing it"? Let's look at a case of electromagnetic induction to make this clearer. Consider a magnet being dropped in a cylindrical tube made up of aluminum, a conducting material. As the magnet falls due to its weight, its field lines are intercepting the edges of the cylinder, creating an induced current in the aluminum material. Due to gravity, the magnet should be accelerating as it falls through the tube, steadily increasing in velocity the lower it drops. However, if the velocity of the magnet is actually measured, and if the cylinder is long enough, we will see the acceleration of the magnet actually decrease.

Fig. 2 - A magnet falling through a conductive tube will experience an upward force due to Lenz's law.

This occurs due to the direction of the current being induced in the tube! As current runs through it, the tube itself generates its own magnetic field. However, this time, the direction of the magnetic field is opposite in direction to that of the magnet falling through the tube. In turn, this causes an opposing force to the weight of the magnet, reducing the net force being applied, and thus decreasing the total acceleration of the magnet. This is a direct consequence of Lenz's law.

Fig. 3 - When the magnet enters the tube, it experiences an upward force due to Lenz's law.

Lenz's Law Formula and Equation

Now that we understand how Lenz's law functions and plays a part in electromagnetic induction, we can also express it in its mathematical form. This equation is given by

\[ \epsilon = - \frac{\mathrm{d} \Phi_{\text{B}}}{\mathrm{d} t} ,\]

where \(\epsilon\) is the induced electromotive force measured in volts \(\mathrm{V}\), \(\Phi_{\text{B}} \) is the magnetic flux from the magnetic field measured in webers \(\mathrm{Wb}\), and \(t\) is the time taken for the change in magnetic flux measured in units of seconds \(\mathrm{s}.\) Here the differential sign indicates that we take the derivative of the expression for the magnetic flux with respect to time.

Looking back to other electromagnetic induction topics, the equation above is actually our equation for electromagnetic induction. The contribution in this equation by Lenz's law is indicated by the negative sign on the right-hand side. This small detail indicates to us that the induced electromotive force is in the opposite direction to that of the changing magnetic flux.

Lenz's Law Right-Hand Rule

We have now established how to calculate Lenz's law from the equation, but how do we determine the direction of the induced electromotive-force, if we know the direction of the changing magnetic flux, and vice-versa? For this, we have a handy trick called the right-hand grip rule.

Assume that the magnet is being dropped with its north pole facing downwards. To use this method, curl the fingers on your right hand as if you are enclosing a tube. The direction of your fingers should follow the direction of the induced current. Then extend out your thumb; the direction your thumb is pointing indicates the north pole of the conductor, as the current has essentially turned the conductor into a magnet. Since we know that the conductor must repel the falling magnet, this means that the north pole of the conductor must be pointing upwards in order to repel the falling magnet with a like pole. Pointing our thumb upwards, we find that the resultant curl of our fingers is in the counter-clockwise direction, which is the direction of the induced current in the tube. This repels the north pole of the magnet from being dropped into the tube, thus creating an opposing force that reduces the acceleration.

The right-hand grip rule can also be used to determine the direction of the magnetic field generated by a current running through a wire. In this case, the thumb points in the direction of the current whilst the direction of your fingers indicate the curvature of the generated magnetic field.

Lenz's Law Examples

Now let's consider some examples where we can apply Lenz's law.

Faraday's Law vs Lenz's Law

Finally, let's briefly cover the differences between Faraday's law and Lenz's law. We can define Faraday's law as the following.

Comparing the two definitions of the two laws, we can see that Faraday's law is concerned with quantifying the amount of electromotive force induced in the conductor, whereas, Lenz's law is concerned with determining the direction of the induced electromotive force. Lenz's law is important as it ensures that the phenomenon of electromagnetic induction obeys the conservation of energy, a fundamental law in any physical system. It is important to note that both of these laws are fundamental in determining electromagnetic induction.