Lenz’s Law AP Physics C Montwood High School R. Casao

The direction of the induced EMF and induced current can be found from Lenz’s law: The polarity of the induced EMF is such that it tends to produce a current that will create a magnetic flux to oppose the change in magnetic flux through the loop. The induced current tends to keep the original flux thru the circuit from changing. Lenz’s law is a consequence of the law of conservation of energy. Consider a bar moving to the right on two parallel rails in the presence of a uniform magnetic field directed into the paper.

As the bar moves to the right, the magnetic flux thru the circuit increases with time since the area of the loop increases. Lenz’s law says that the induced current must be in a direction such that the flux it produces opposes the change in the external magnetic flux. Since the flux due to the external field is increasing into the page, the induced current, if it is to oppose the change in flux, must produce a flux that is out of the page.

The induced current must be counterclockwise when the bar moves to the right to give a counteracting flux out of the page in the region inside the loop. You can verify this with the right hand rule: the right thumb points up so that the fingers curl out of the page.

If the bar is moving to the left, the magnetic flux thru the loop decreases with time. Since the flux is into the page, the induced current has to be clockwise to produce a flux into the page inside the loop. Using the right hand rule: the thumb of the right hand points downward so that the fingers of the right hand curl down into the page to maintain the flux inside the loop.

In terms of energy conservation: the current was previously shown to be counterclockwise in the loop. If the current was clockwise in the loop, the direction of the magnetic force on the sliding bar would be to the right. A force to the right would accelerate the rod and increase its velocity. An increase in velocity would cause the area of the loop to increase more rapidly, increasing the induced current,... The system would acquire energy with no additional energy input; violating the law of conservation of energy.

Consider a bar magnet moved to the right toward a stationary loop of wire. As the magnet moves to the right toward the stationary loop, the magnetic flux thru the loop increases with time. To counteract the increase in flux to the right, the induced current produces a flux to the left and the induced current in the loop is in the direction shown in the figures.

Notice that the magnetic field lines associated with the induced current (figure on the right) oppose the motion of the magnet. The left face of the current loop is a north pole and the right face is a south pole.

If the magnet were moving to the left, the flux thru the loop would be decreasing with time. To counteract the decrease in flux, an induced current in the loop would be in a direction such as to set up a field thru the loop directed left to right in an effort to maintain a constant number of flux lines. The induced current in the loop would be directed as shown in the figures.

Notice that the magnetic field lines associated with the induced current (figure on the right) are in the direction of magnetic field lines associated with the motion of the magnet. The left face of the current loop is a south pole and the right face is a north pole.

Application of Lenz’s Law A coil of wire is placed near an electromagnet as shown in the figure. Find the direction of the induced current in the coil at the instant the switch is closed. When the switch is closed, the situation changes from a condition in which no lines of flux pass thru the coil to one in which lines of flux pass thru the coil in the direction shown.

To counteract the increase in the number of lines passing thru the coil, the coil must set up a magnetic field from left to right. The left to right direction requires and induced current directed from top to bottom as shown in the figure. Direction of magnetic field thru electromagnet found by curling fingers of right hand in direction of current I; thumb points to left (N pole). Number of magnetic field lines passing thru coil increases from 0 to N; induced current I produces magnetic field lines that will point from left to right to oppose change in flux.

Curl fingers of right hand in direction of left to right thru coil and thumb points downward to give you the direction of the induced current. Find the direction of the induced current in the coil after the switch has been closed for several seconds. After the switch has been closed for several seconds and the steady-state current is established, there is no change in the number of magnetic field lines passing thru the loop, therefore, there is no induced current in the loop.

Find the induced current in the coil when the switch is opened. Opening the switch causes the magnetic field to change from a condition in which flux lines pass thru the coil from right to left to a condition in which there is zero flux. To counteract the decrease in the number of lines passing thru the coil, the coil must set up a magnetic field from right to left.

The right to left direction requires and induced current directed from bottom to top as shown in the figure. Direction of magnetic field thru electromagnet found by curling fingers of right hand in direction of current I; thumb points to left (N pole). Number of magnetic field lines passing thru coil decreases from N to 0; induced current I produces magnetic field lines that will point from right to left to oppose change in flux.

Curl fingers of right hand in direction of right to left thru coil and thumb points upward to give you the direction of the induced current.

A Loop Moving Thru a B Field A rectangular loop of dimensions l and w and resistance R moves with constant speed v to the right. It continues to move with this speed thru a region containing a uniform magnetic field B directed into the paper and extending a distance 3·w. Plot the flux, the induced EMF, and the external force acting on the loop as a function of the position of the loop in the field.

Before the loop enters the magnetic field, the flux is 0. As it enters the field, the flux increases linearly with position as the area of the loop within the magnetic field increases to its maximum at w. As the entire loop passes thru the magnetic field from w to 3·w, the flux thru the loop does not change and remains at its maximum value.

As the loop reaches the right edge of the magnetic field and begins to pass out of the field, the flux decreases linearly to 0 as the area of the loop that is in the magnetic field decreases from a maximum value to 0.

Before the loop enters the field, there is no induced EMF since there is no flux passing thru the loop. As the right side of the loop enters the magnetic field, the flux thru the loop begins to increase, inducing a current and EMF in the loop. The induced magnetic field is up out of the page to oppose the change in the flux down into the page.

Curling the fingers of the right hand upward out of the page, the thumb indicates that the current (and EMF) direction would be counterclockwise (-B· l ·v). As the entire loop passes thru the magnetic field and the flux remains constant, there is no induced current or EMF in the loop.

As the right side of the loop exits the magnetic field, the flux thru the loop begins to decrease, inducing a current and EMF in the loop. The induced magnetic field is down and into the page to oppose the decrease in the flux down into the page. Curling the fingers of the right hand downward into the page, the thumb indicates that the current (and EMF) direction would be clockwise (B· l ·v).

Once the area of the loop is entirely out of the magnetic field, there is no change in the flux thru the loop, therefore, there is no induced current or EMF in the loop and the EMF is again 0.

When the loop is not in the magnetic field, no magnetic force acts on the loop. When the right side of the loop enters the field, the external force needed to maintain constant speed must be equal and opposite to the magnetic force exerted on the right side of the loop. The left side of the loop is not in the magnetic field, so no force is exerted on the left side of the loop.

Using F = I· l ·B, the direction of the force on the right side points to the left, therefore, the external force to maintain constant velocity is directed to the right. When the loop is entirely in the magnetic field, the flux thru the loop does not change and no current or EMF is induced in the loop, therefore, no external force is needed to maintain the constant velocity of the loop.

When the right side of the loop exits the field, the external force needed to maintain constant speed must be equal and opposite to the magnetic force exerted on the left side of the loop. The right side is no longer in the field, so no force acts on it. Using F = I· l ·B, the direction of the force on the left side points to the left, therefore, the external force to maintain constant velocity is directed to the right.