# Lenz's Law and the Law of Conservation of Energy

This is part of the HSC Physics course under the topic Electromagnetic Induction.

### HSC Physics Syllabus

• analyse qualitatively and quantitatively, with reference to energy transfers and transformations, examples of Faraday's Law and Lenz's Law \varepsilon = -N (∆\phi)/(∆t), including but not limited to: (ACSPH081, ACSPH110)
– the generation of an electromotive force (emf) and evidence for Lenz's Law produced by the relative movement between a magnet, straight conductors, metal plates and solenoids
– the generation of an emf produced by the relative movement or changes in current in one solenoid in the vicinity of another solenoid

### Lenz's Law and The Law of Conservation of Energy

This video will analyse qualitatively and quantitatively, with reference to energy transfers and transformations, examples of Lenz's Law. It will also explain how it factors into the equation \varepsilon = -N (∆\phi)/(∆t).

### What is Lenz's Law?

Lenz's law states that the direction of the induced current is such that the magnetic flux produced will oppose the change in flux which induced in the EMF in the first place. Consequently, the induced current's own magnetic field will interacts with the external magnetic field so as to produce a 'resisting' force.

Lenz’s law comes hand in hand with Faraday's law because the current described in Lenz's law is due to the induced EMF described in Faraday's law.

However, Lenz's law is only applicable if a current is produced. For example, if EMF is induced in an open circuit where a current cannot flow, Lenz's law is not applicable and a resisting force will not be present.

The best way to understand and master Lenz's law is to learn the examples of Lenz's law (see below).

### Identifying Flux Change & Electromagnetic Induction

Since Lenz's law cannot occur without electromagnetic induction (Faraday's law), the best way to identify when Lenz's law is applicable is to recognise when a conductor experiences changes in magnetic flux.

A common cause of flux change is a relative movement between a conductor and a magnetic field. For example:

• moving a magnet in the vicinity of a conductor or
• moving a conductor within a non-uniform magnetic field

### Straight Conductors

When a straight conductor experiences changes in magnetic flux, a current is induced such that its accompanying magnetic field would oppose the change. The induced current would be in the direction so as to try to restore the magnetic flux it was originally experiencing.

Let's explain this further using an example. Suppose a straight conductor is moved to the right into a uniform magnetic field directed into the page/screen as shown below.

This movement causes the straight conductor to experience an increase in flux. By Lenz's law, the direction of induced current would be such that the result force would be directed to the left (opposing the movement of the conductor).

Using the right hand palm rule for a current-carrying conductor in magnetic fields (motor effect):

• Palm faces the left as this is where the opposing force would be directed to
• Fingers point into the screen/page to represent the direction of the magnetic field
• Thumb points up the screen/page to indicate the direction of induced current.

Therefore, the induced current will flow upwards.

Another way to understand why the induced current flows in such a direction is to think of the current as moving electrons. When the straight conductor is moved into the magnetic field (or even within it), mobile electrons will experience a magnetic force (F = qvB\sin \theta).

In the diagram above, using the right hand palm rule:

• Thumb points to the left because negatively charged electrons are moved to the right due to movement of the conductor
• Fingers point into the page/screen to represent the direction of the magnetic field.
• Palm would be facing down the page/screen to indicate the force acting on these electrons. As a result of the force, these electrons will move to the bottom end of the conductor. This means conventional current will flow towards the top end of the conductor – consistent with Lenz's law.

The opposing force will decelerate straight conductor, decreasing its kinetic energy. The kinetic energy is transformed into electrical energy, justifying the presence of the induced current.

### How Does the Law of Conservation of Energy Justify Lenz's Law?

Induced current is a form of electrical energy. This energy must have originated from somewhere due to Law of Conservation of Energy. That is, the kinetic energy of the relative motion between a conductor and external magnetic field must have decreased and been transformed into electrical energy. This is why it makes sense for there to be a resisting force, slowing down the relative motion.

An alternative justification is to consider a counter-example. Let's assume Lenz's Law was not true and that the induced current does not act to oppose the change in flux but instead acts to promote it.

If this was the case, when a straight conductor is moved into a magnetic field, the resultant force would further accelerate the conductor in order to further increase the magnetic flux it was experiencing. The acceleration not only increases the conductor's kinetic energy but also leads to a greater rate of change in flux and thus higher magnitude of induced current (electrical energy). The increase in both kinetic and electrical energy violates energy conservation as energy has been created out of nowhere.

### Coils of Wire (Solenoids)

Moving a permanent magnet in and out of coiled wires induces current. The induced current generates its own magnetic field which in turn is orientated such that it opposes the external magnetic field of the magnet.

When induced current flows through coiled wires, it essential acts as a solenoid. The orientation of a solenoid's magnetic field is determined by the right-hand rule:

Suppose a permanent magnet is positioned with its south pole facing the coiled wire. This magnet is then moved towards and away from the wire at the same speed.

When the magnet is moved towards the coiled wire, it experiences an increase in magnetic flux. As a result, a magnetic field is created to repel the incoming magnet. In order to produce this repulsive force, the induced current must flow in the direction shown above such that the south pole of its magnetic field is on the left side.

When the magnet is moved away from the solenoid, the solenoid experiences a decrease in magnetic flux. As a result, a magnetic field is created to attract the incoming magnet. To produce this attractive force, the induced current must flow a direction such that the north pole of its magnetic field is on the left side.

It is important to understand that the direction of induced current depends on the relative motion between the magnetic field (magnet) and the conductor (coiled wires). The directions of induced current are opposite in the two cases described above because the nature of flux change is difference (first case was due to an increase in flux whereas the second case was due to a decrease in flux).

When the magnet is fully inside the coiled wires, no changes in magnetic flux are experienced. Consequently, Lenz's law does not apply.

Some more examples are shown below.

Observations:

• In a), the magnet is at rest so there is no deflection in the ammeter, so the needle of ammeter is at centre or zero position.
• In b), when the magnet is moved toward the coil, the needle of ammeter deflects in one direction.
• In c), when the magnet is held stationary at that position, the needle of ammeter returns back to zero position.
• In d), when the magnet is moved away from the coil, there is some deflection in the needle but in opposite direction.

### Metal Plates & Metal Sheets

The induced current in a metal plate as a result of Faraday and Lenz’s law is called eddy current, which are circular closed currents. Eddy currents are only induced in conductors of large surface area, usually metal plates, cylindrical pipes made of conducting material.

The direction of the induced eddy current (either clockwise or anti-clockwise) can be determined by the right-hand grip rule). Thumb points in the direction of the magnetic flux the metal plate desires to restore. The direction the four fingers curl in is the direction of the eddy current.

For example, in the diagram above, a square metal plate is moved out of the magnetic field going into the page/screen. As a result, the metal plate experiences a decrease in magnetic flux. Specifically, a decrease of flux directed into the page. By Lenz's law, the induced current wants to increase the magnitude of flux directed into the page/screen.

Let's apply the right-hand grip rule:

• Thumb points into the page to represent the magnetic field.
• Fingers are curled in a clockwise manner – this is the direction of the resultant eddy current.

### Induction by Moving One Solenoid in the Vicinity of Another Solenoid

A solenoid produces a magnetic field when current flows through it.

Suppose we have two solenoids: one solenoid (left) is connected to an ammeter or galvanometer (both measure current), and another solenoid (right) is corrected to a switch and battery.

Closing the switch in the right solenoid will cause a current to flow, resulting in a magnetic field. Vice versa, when the switch is opened, this magnetic field disappears.

Therefore, closing and opening the switch in the right solenoid will cause the left solenoid to experience an increase and decrease in magnetic flux respectively.

In both cases, by Faraday’s law, an EMF is induced, which results in current (as the left solenoid is a closed circuit).

By Lenz’s law, the direction of current is different when the switch is closed compared with when it’s open. This is shown by the pointer being deflected in opposite direction in the galvanometer.

• When the switch is closed (increase in flux), the induced current in the left solenoid will flow in a direction such that it tries to repel the left solenoid away in order to reduced the flux.
• When the switch is opened (decrease in flux), the induced current in the left solenoid will flow in a direction such that it tries to attract the right solenoid in order to increase the flux.

The right hand grip rule can be used to determine the direction of induced current required to achieve this.

Shortly after the switch has been closed or opened, the induced current in the left solenoid quickly disappears. This is because the changes in magnetic flux only occur at the instance when the switch is closed or opened. When the magnetic flux experienced by the left solenoid becomes constant, no induction occurs.

This way of transmitting electrical energy from one conductor to another is called flux linkage which is a concept that underpins the operation of transformers.

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