When traveling through an airport, you're usually asked to place your belongings through a scanner and then asked to walk into a large metal detector. You may have asked yourself, how are these large metal gates even checking anything, and why do they beep if I've accidentally left my phone in my pocket? The key component of these gates is that they produce an alternating magnetic field as you walk through them. Therefore, any metal item you may have on you, interacts with the field, generating its own opposing, magnetic field through the laws of electromagnetic induction. Thus the scanner is able to detect this change in the magnetic field, resulting in the alarm going off, and you're phone being confiscated from you! The laws governing this phenomenon of electromagnetic induction are Lenz's law and Faraday's law, keep reading to learn more!
Fig. 1 - Airport metal detectors use electromagnetic induction to scan for metallic objects being carried by passengers.
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.
Electromagnetic induction is the formation of an electromotive force, or EMF, due to the motion of an electromagnetic field near an electrical conductor.
We can also define EMF as the following.
The electromotive force (EMF) is the potential difference between two points imparted by an energy source. In other words, the EMF is the energy imparted per coulomb of charge by an energy source.
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.
Lenz's law states that the direction of an induced current will always be flowing in such a direction that it will oppose the motion causing it.
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.
Referring to the figure below, we see a magnet moving alongside a stationary tube made out of a conducting material.
Fig. 4 - The changing magnetic flux due to the moving magnet will induce an EMF in the conducting tube.
Due to the movement of the magnet, the conductive tube experiences a change in magnetic flux over time. We can express this magnetic flux as
\[ \Phi_{\text{B}} = at^2 + bt ,\]
where \(a\) is a constant given by a value of \( - 12 \, \mathrm{\frac{Wb}{s^2}} \) and \(b\) is a constant given by a value of \(5.0 \, \mathrm{\frac{Wb}{s}}\).
Given this information, solve for
- The induced EMF in the conducting tube at time \(t = 1.5 \, \mathrm{s}\).
- The reading on the ammeter given that the resistance of the conducting tube is \(R = 1.5 \, \Omega \).
1. Firstly, using the given expression for the magnetic flux, we calculate the time derivative as
\[ \frac{\mathrm{d} \Phi_{\text{B}}}{\mathrm{d} t} = 2at + b .\]
Thus we now know our equation for the induced EMF is
\[ \epsilon = - \frac{\mathrm{d} \Phi_{\text{B}}}{\mathrm{d} t} = -2at - b .\]
Substituting in the values of the constant and for time \(t = 1.5 \, \mathrm{s} \) results in
\[ \begin{align} \epsilon &= - (2 \times -12 \, \mathrm{\frac{Wb}{s^2}} \times 1.5 \, \mathrm{s} ) - ( 5.0 \, \mathrm{\frac{Wb}{s}} ) \\ \epsilon &= 31 \, \mathrm{\frac{Wb}{s}} \\ \epsilon &= 31 \, \mathrm{V} , \end{align} \]
where we have used the fact that \( 1 \, \mathrm{\frac{Wb}{s}} = 1 \, \mathrm{V} \).
2. Given that we have the induced EMF, we can now calculate the current running through the circuit using Ohm's law. Substituting the values we have calculated results in
\[ \begin{align} I &= \frac{V}{R}, \\ I &= \frac{31 \, \mathrm{V}}{1.5 \, \Omega}, \\ I &= 21 \, \mathrm{A} . \end{align} \]
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.
Faraday's law states that the induced electromotive force in a conductor, due to electromagnetic induction, is proportional to the rate of change of magnetic flux.
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.
Lenz's Law - Key takeaways
- Lenz's law states that the direction of an induced current will always be in the direction to oppose the motion causing it.
- Both Faraday's law and Lenz's law are crucial to the phenomenon of electromagnetic induction.
- We can define Lenz's law mathematically as \( \epsilon = - \frac{\mathrm{d} \Phi_{\text{B}}}{\mathrm{d} t} \), where the negative sign on the right-hand side is the contribution due to Lenz's law.
- The direction of the opposing force due to Lenz's law can be determined using the right-hand grip rule.
- Faraday's law is concerned with the magnitude of the induced electromotive force, whilst Lenz's law is concerned with the direction of the induced electromotive force.
References
- Fig. 1 - Airport security, Wikimedia Commons (https://commons.wikimedia.org/wiki/File:FEMA_-_37752_-_Residents_at_the_airport_preparing_to_leave_Louisiana.jpg) Public Domain.
- Fig. 2 - Magnet falling through conductive tube, StudySmarter Originals.
- Fig. 3 - Magnet falling inside tube, StudySmarter Originals.
- Fig. 4 - Right-hand grip rule, StudySmarter Originals.
- Fig. 5 - Moving magnet next to tube, StudySmarter Originals.
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