Physics For Scientists And Engineers 6E - part 245

SECTION 31.3 • Lenz’s Law

977

31.3 Lenz’s Law

Faraday’s law (Eq. 31.1) indicates that the induced emf and the change in flux have
opposite algebraic signs. This has a very real physical interpretation that has come to
be known as

Lenz’s law

The induced current in a loop is in the direction that creates a magnetic field that
opposes the change in magnetic flux through the area enclosed by the loop.

Developed by the German physicist Heinrich Lenz (1804–1865).

horizontal direction gives

From Equation 31.6, we know that I " B!v/R, and so we
can write this expression as

Integrating this equation using the initial condition that
v " v

at t " 0, we find that

where the constant - " mR/B

. From this result, we see

that the velocity can be expressed in the exponential form

(1)

To finalize the problem, note that this expression for v indi-
cates that the velocity of the bar decreases with time under
the action of the magnetic retarding force, as we expect
from our conceptualization of the problem.

(B) The text of part (B) immediately categorizes this as a

problem in energy conservation. Consider the sliding bar as
one system possessing kinetic energy, which decreases because
energy is transferring out of the system by electrical transmis-
sion through the rails. The resistor is another system possess-
ing internal energy, which rises because energy is transferring
into this system. Because energy is not leaving the combination
of two systems, the rate of energy transfer out of the bar equals
the rate of energy transfer into the resistor. Thus,

resistor

" %

bar

where the negative sign is necessary because energy is
leaving the bar and "

bar

is a negative number. Substituting

for the electrical power delivered to the resistor and the

v " v

t/-

" %

t " %

" %

ma " m

" %

I!B

time rate of change of kinetic energy for the bar, we have

Using Equation 31.6 for the current and carrying out the
derivative, we find

Rearranging terms gives

To finalize this part of the problem, note that this is the
same expression that we obtained in part (A).

What If?

Suppose you wished to increase the distance

through which the bar moves between the time when it is ini-
tially projected and the time when it essentially comes to rest.
You can do this by changing one of three variables: v

, R, or B,

by a factor of 2 or . Which variable should you change in order
to maximize the distance, and would you double it or halve it?

Answer Increasing v

would make the bar move farther.

Increasing R would decrease the current and, therefore, the
magnetic force, making the bar move farther. Decreasing B
would decrease the magnetic force and make the bar move
farther. But which is most effective?

We use Equation (1) to find the distance that the bar

moves by integration:

From this expression, we see that doubling v

or R will

double the distance. But changing B by a factor of causes
the distance to be four times as great!

" %v

(0 % 1) " v

- "

x "

t/-

dt " %v

t/-

v "

t/-

" %

R " %

(

)

At the Interactive Worked Example link at http://www.pse6.com, you can study the motion of the bar after it is released.

Lenz’s law

That is, the induced current tends to keep the original magnetic flux through the
circuit from changing. We shall show that this law is a consequence of the law of
conservation of energy.

To understand Lenz’s law, let us return to the example of a bar moving to the

right on  two  parallel  rails  in  the  presence  of  a  uniform  magnetic  field  (the  external
magnetic  field,  Fig.  31.13a.)  As  the  bar  moves  to  the  right,  the  magnetic  flux
through the  area  enclosed  by  the  circuit  increases  with  time  because  the  area
increases.  Lenz’s  law  states  that  the  induced  current  must  be  directed  so  that  the
magnetic  field  it  produces  opposes  the  change  in  the  external  magnetic  flux.
Because the magnetic flux due to an external field directed into the page is increas-
ing, the  induced  current,  if  it  is  to  oppose  this  change,  must  produce  a  field
directed out  of  the  page.  Hence,  the  induced  current  must  be  directed  counter-
clockwise  when  the  bar  moves  to  the  right.  (Use  the  right-hand  rule  to  verify  this
direction.)  If  the  bar  is  moving  to  the  left,  as  in  Figure  31.13b,  the  external  mag-
netic flux  through  the  area  enclosed  by  the  loop  decreases  with  time.  Because  the
field is  directed  into  the  page,  the  direction  of  the  induced  current  must  be
clockwise if it is to produce a field that also is directed into the page. In either case,
the induced  current  tends  to  maintain  the  original  flux  through  the  area  enclosed
by the current loop.

Let us examine this situation using energy considerations. Suppose that the bar is

given a slight push to the right. In the preceding analysis, we found that this motion
sets up a counterclockwise current in the loop. What happens if we assume that the

978

CHAPTE R 31 • Faraday’s Law

Figure 31.13 (a) As the

conducting bar slides on the two

fixed conducting rails, the

magnetic flux due to the external

magnetic field into the page

through the area enclosed by the

loop increases in time. By Lenz’s

law, the induced current must be

counterclockwise so as to produce

a counteracting magnetic field

directed out of the page. (b) When

the bar moves to the left, the

induced current must be clockwise.

Why?

(a)

(b)

Figure 31.14 (a) When the magnet is moved toward the stationary conducting

loop, a current is induced in the direction shown. The magnetic field lines shown

are those due to the bar magnet. (b) This induced current produces its own

magnetic field directed to the left that counteracts the increasing external flux. The

magnetic field lines shown are those due to the induced current in the ring.

is induced in the direction shown. The magnetic field lines shown are those due to

the bar magnet. (d) This induced current produces a magnetic field directed to the

right and so counteracts the decreasing external flux. The magnetic field lines

shown are those due to the induced current in the ring.

Example

(b)

(a)

(d)

(c)

current is clockwise, such that the direction of the magnetic force exerted on the bar is
to the right? This force would accelerate the rod and increase its velocity. This, in turn,
would cause the area enclosed by the loop to increase more rapidly; this would result
in an increase in the induced current, which would cause an increase in the force,
which would produce an increase in the current, and so on. In effect, the system would
acquire energy with no input of energy. This is clearly inconsistent with all experience
and violates the law of conservation of energy. Thus, we are forced to conclude that the
current must be counterclockwise.

Let us consider another situation, one in which a bar magnet moves toward a

stationary metal loop, as in Figure 31.14a. As the magnet moves to the right toward the
loop, the external magnetic flux through the loop increases with time. To counteract
this increase in flux due to a field toward the right, the induced current produces its
own magnetic field to the left, as illustrated in Figure 31.14b; hence, the induced
current is in the direction shown. Knowing that like magnetic poles repel each other,
we conclude that the left face of the current loop acts like a north pole and that the
right face acts like a south pole.

If the magnet moves to the left, as in Figure 31.14c, its flux through the area

enclosed by the loop decreases in time. Now the induced current in the loop is in the
direction shown in Figure 31.14d because this current direction produces a magnetic
field in the same direction as the external field. In this case, the left face of the loop is
a south pole and the right face is a north pole.

SECTION 31.3 • Lenz’s Law

979

Quick Quiz 31.7

Figure 31.15 shows a magnet being moved in the vicinity

of a solenoid connected to a sensitive ammeter. The south pole of the magnet is the
pole nearest the solenoid, and the ammeter indicates a clockwise (viewed from
above) current in the solenoid. Is the person (a) inserting the magnet or (b) pulling
it out?

Quick Quiz 31.8

Figure 31.16 shows a circular loop of wire being dropped

toward a wire carrying a current to the left. The direction of the induced current in
the loop of wire is (a) clockwise (b) counterclockwise (c) zero (d) impossible to
determine.

Figure 31.15 (Quick Quiz 31.7)

Richard

Megna/Fundamental

Photographs

Figure 31.16 (Quick Quiz 31.8)

980

CHAPTE R 31 • Faraday’s Law

Conceptual Example 31.6 Application of Lenz’s Law

A metal ring is placed near a solenoid, as shown in Figure
31.17a. Find the direction of the induced current in the ring

(A)

at the instant the switch in the circuit containing the

solenoid is thrown closed,

(B)

after the switch has been closed for several seconds, and

(C)

at the instant the switch is thrown open.

Solution (A) At the instant the switch is thrown closed, the
situation changes from one in which no magnetic flux exists
in the ring to one in which flux exists and the magnetic field
is to the left as shown in Figure 31.17b. To counteract this
change in the flux, the current induced in the ring must set
up a magnetic field directed from left to right in Figure
31.17b. This requires a current directed as shown.

(B) After the switch has been closed for several seconds,

no change in the magnetic flux through the loop occurs;
hence, the induced current in the ring is zero.

which magnetic flux exists in the ring to one in which there is
no magnetic flux. The direction of the induced current is as
shown in Figure 31.17c because current in this direction

produces a magnetic field that is directed right to left and so
counteracts the decrease in the flux produced by the solenoid.

(c)

(a)

(b)

Switch

Figure 31.17 (Example 31.6) A current is induced in a metal

ring near a solenoid when the switch is opened or thrown

closed.

Conceptual Example 31.7 A Loop Moving Through a Magnetic Field

A rectangular metallic loop of dimensions ! and w and
resistance R moves with constant speed v to the right, as in
Figure 31.18a. The loop passes through a uniform magnetic
field

B directed into the page and extending a distance 3w

along the x axis. Defining x as the position of the right side
of the loop along the x axis, plot as functions of x

(A)

the magnetic flux through the area enclosed by the loop,

(B)

the induced motional emf, and

(C)

the external applied force necessary to counter the

magnetic force and keep v constant.

Solution (A) Figure 31.18b shows the flux through the
area enclosed by the loop as a function of x. Before the loop
enters the field, the flux is zero. As the loop enters the field,
the flux increases linearly with position until the left edge of

the loop is just inside the field. Finally, the flux through the
loop decreases linearly to zero as the loop leaves the field.

(B) Before the loop enters the field, no motional emf is

induced in it because no field is present (Fig. 31.18c). As the
right  side  of  the  loop  enters  the  field,  the  magnetic  flux
directed into the page increases. Hence, according to Lenz’s
law, the induced current is counterclockwise because it must
produce its own magnetic field directed out of the page. The
motional emf % B!v (from Eq. 31.5) arises from the magnetic
force  experienced  by  charges  in  the  right  side  of  the  loop.
When the loop is entirely in the field, the change in magnetic
flux  is  zero,  and  hence  the  motional  emf  vanishes.  This
happens  because,  once  the  left  side  of  the  loop  enters  the
field, the motional emf induced in it cancels the motional emf
present in the right side of the loop. As the right side of the
loop leaves the field, the flux begins to decrease, a clockwise

(d)

3w 4w

(b)

B !w

(a)

(c)

B !v

– B !v

Figure 31.18 (Conceptual Example 31.7) (a) A

conducting rectangular loop of width w and length !

moving with a velocity v through a uniform magnetic field

extending a distance 3w. (b) Magnetic flux through the

area enclosed by the loop as a function of loop position.

(d) Applied force required for constant velocity as a

function of loop position.

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