Download 03-10--L5-Magnetic Fields and Forces

Physics E-1ax
March 10, 2015
Magnetic Fields and Forces
For permanent magnets (e.g. bar magnets), like poles repel; unlike poles attract.
Although there are some similarities between electric and magnetic fields, there is one
crucial difference:
Magnetic field lines never begin or end anywhere; they go in closed loops
So a magnetic “pole” is really just a place where the field lines bunch together.
A bar magnet is like a magnetic dipole, and feels a torque in a uniform magnetic field.
Magnetic fields can be created by currents. The field forms “loops” around the current,
with direction determined by the right hand rule, and a magnitude of:
µ I
B= 0
2π r
Magnetic fields exert forces on moving charges, such that the force is always
perpendicular to the direction of motion of the charge. Likewise, magnetic fields exert
forces on a current-carrying wire (because the wire contains moving charges).
A loop of current creates a magnetic dipole.
If the area of the loop is A, and the current

is i, the magnetic dipole moment
is
,
and
the energy of a magnetic dipole in a
µ
=
i
A
 
uniform field is U mag = −µ ⋅ B .


 
The magnetic flux
 of the field B through an area A is: Φ B = B ⋅ A = BA cosθ , where
the area vector A determines the orientation around the edge according to the right-hand
rule. Usually we’ll be interested in the magnetic flux through a loop of wire.
You can change the magnetic flux by changing the area of a loop, changing the
magnitude of the magnetic field, or changing the angle between the magnetic field
vector and the area vector.
Faraday’s Law of Induction says that a changing magnetic flux will induce an EMF in
a loop of wire. This induced EMF will cause current to flow in the wire. Faraday’s law
is the basis for understanding how electric power is generated, and it explains (in part)
why your household power uses alternating current (AC).
•
Learning objectives: After this lecture, you will be able to…
1.
Identify the poles of a magnet, and sketch the magnetic field lines created by a permanent
magnet (like a bar magnet).
2.
Construct an analogy between magnetic poles and electric charges, and explain the
limitations of that analogy.
3.
Construct an analogy between magnetic field lines and electric field lines, and explain the
limitations of that analogy.
4.
Predict the most stable configuration of a magnetic dipole in uniform magnetic field, and
explain why two magnetic dipoles (bar magnets) attract or repel one another.
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Physics E-1ax
March 10, 2015
5.
Calculate and draw the magnetic field created by a current in a long straight wire, using
the right-hand rule to predict the correct direction of the field.
6.
Calculate the magnitude and direction of the magnetic force on a moving electric charge,
using the cross product and the right-hand rule.
7.
Calculate the magnitude and direction of the magnetic force on a long straight wire
carrying an electric current.
8.
Predict the direction of the magnetic force between two long straight wires, both with
currents flowing in them.
9.
Identify the magnitude and direction of the magnetic dipole moment created by a circular
loop of current.
10.
Calculate the energy of a magnetic dipole placed in an external magnetic field.
11.
Define the magnetic flux through a loop of wire, and calculate the flux for a given
magnetic field and loop (including the sign of the flux).
12.
Describe what you can do to change the magnetic flux through a loop, and decide
whether the magnitude of the flux gets larger or smaller for each type of change.
13.
Explain Faraday’s Law of Induction and how it relates changes in magnetic flux with
induced current in a loop.
14.
Use Lenz’s Law to determine the direction of the induced current in a loop.
15.
Calculate the magnitude and direction of the induced current in a loop of wire, given the
change in the magnetic flux and the resistance of the wire, using both Faraday’s Law and
Lenz’s Law.
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Physics E-1ax
March 10, 2015
Am I getting it?
•
For the circuit shown below, indicate how the LED will respond to the voltages at the
inputs A and B. (In other words, does the light turn on or off?)
+5V
+5V
A
A
(V)
B
(V)
0
0
0
+5
+5
0
+5
+5
B
0V
0V
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LED
(on/off)
Physics E-1ax
March 10, 2015
Magnets and Magnetic Fields
•
You’re all familiar with the basic properties of magnets with north and south poles:
•
On a compass, the north end of the compass needle points towards the north pole of the
Earth. That means that the Earth’s north pole is actually…
•
There are some similarities between magnetic poles and electric charges:
•
However, single magnetic poles do not exist.
What happens if you take a bar magnet and cut it
in half?
•

Just as charges create electric fields, permanent magnets create the magnetic field, B :
There is a key difference between these two kinds of fields, however:
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Physics E-1ax
March 10, 2015
Activity 1: Permanent Magnets: Fields and Forces
1.
A typical kind of permanent magnet is a bar magnet, which acts like a large magnetic
dipole. Suppose you place a bar magnet in the uniform magnetic field shown below.
What will be the most stable orientation of the bar magnet?
•
You know that two bar magnets will attract or repel depending on their orientation. But
why? Let’s consider first the case of attraction. The magnet on the left is creating a
magnetic field; the magnet on the right will experience a force.
2.
Consider the force on the poles on the magnet on the right. What will be the force
(magnitude/direction) on the N pole? What will be the force (magnitude/direction) on the
S pole? What will be the net force? (Hint: N is analogous to (+); consider field strength)
3.
Now make the same argument to find the net force on the magnet in the configuration
shown below.
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March 10, 2015
Creating Magnetic Fields from Currents
•
Instead of using a permanent magnet, we can create a magnetic field from an electric
current. Let’s see a demonstration using a long straight conducting wire. In what
direction does the magnetic field point?
•
If we view the current pointing out of the page, the magnetic field lines look like this:
We can remember the direction that it “curls” using a right-hand rule:
•
The magnitude of the magnetic field around a long straight wire is given by:
B=
µ0 I
2π r
with µ0 = 4π × 10–7 T·m/A. The SI unit of magnetic field is T = Tesla.
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Physics E-1ax
March 10, 2015
Activity 2: Magnetic Force on a Moving Charge
•
Just as moving charges (currents) can create a magnetic field, a magnetic field also exerts
a force on a moving charge. Let’s see a demonstration.
1.
What direction are the charges (electrons) moving?
2.
Fill in the table below to account for your observations:
Magnetic field direction:
Direction of the force on the
electrons:
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A wide variety of these kinds of experiments have shown that:
- There is no magnetic force on a static charge—the charge must be moving.
- The force is proportional to:
the amount of charge q (including the sign!)
the speed of the charge v
the strength of the magnetic field B.


- The force is perpendicular to both v and B .


- The force is proportional to the sine of the angle between v and B .
•
So the magnitude of the magnetic force on a moving charge is:
+q
θ
(out of page) FB = qvBsin θ
But what about the direction?
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March 10, 2015
Cross Product and Right-Hand Rule
•
•
We will need a new kind of vector product,
called the cross product. Consider any two

vectors a and b . Define a vector c , the cross product, as follows:

The cross product c :
- is a vector
  
- has magnitude c = a b sin θ


- has direction perpendicular to both a and b
- has orientation given by the right hand rule
   
- is NOT commutative: a × b ≠ b × a
•
  
To find the direction of the cross product c = a × b , use your right hand:
- Take your right hand

- Hold out your hand straight, pointing in the direction of a .

- Bend your fingers so your fingertips point in the direction of b .
  
- Stick out your thumb, which will point in the direction of c = a × b .
•
There’s a great book about the right hand rule…
•
Now we can write a concise expression for the
magnetic force on a moving charge:

 
FB = qv × B
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Physics E-1ax
March 10, 2015
Am I getting it?
Magnetic Force on Charges and Currents
•
What is the direction of the magnetic force on each of the following moving charges?
1.
2.
3.
4.
5.
6.
•
Are these results consistent with what we saw in the demonstration with the electrons?
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Physics E-1ax
March 10, 2015
Activity 3: Make it Jump!
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If charges are moving through a wire (i.e. a current), then we can express the total
magnetic force on the wire as:

 
FB = iL × B
1.
I’m going to connect the long wires (shown below) to the truck battery. What will
happen to the wires?
• What is the direction of the current in the wire?
• Consider the left part of the long wire. What is the magnetic field created by that wire
(everywhere in space)? Draw it below… (Hint: you’ll need to use dots and crosses…)
• Now consider the right wire. What is the magnetic force on the right wire due to the
magnetic field created by the left wire? (Hint: see above…)
• What about the force of the right wire on the left wire? (Hint: Newton…)
• So what will happen to the wires?
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Physics E-1ax
March 10, 2015
Activity 4: Magnetic Dipoles
What would be the magnetic field created by a loop of current? Let’s see a demo…
•
We can define a magnetic dipole moment for a current loop as:
 
µ = iA
A
•
are
a
vec
tor
How do we define the area vector in this equation?
current loop
1.
!
!
The magnetic energy of a magnetic dipole µ in a magnetic field B is:
 
U mag = −µ ⋅ B
In the field below, draw three dipoles (bar magnets): one with the lowest possible energy,
one with the highest possible energy, and one with some intermediate energy.
2.
The magnetic dipole vector points from (circle one):
S to N
!
!
Bonus! What is the electric potential energy of an electric dipole p in an electric field E ?
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N to S
Physics E-1ax
March 10, 2015
Magnetic Flux
We need to define a quantity called magnetic flux:
Given a small area A and a magnetic field B, we can
define the flux as:
 
Φ B = B ⋅ A = BA cosθ
are
a
vec
tor
A
•
direction around perimeter
•
It will be important to define a direction around the perimeter based on the choice of
direction for the area vector A using the right hand rule.
•
Can the magnetic flux be negative? Yes! It depends on your choice of the area vector:
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March 10, 2015
Faraday’s Observations
•
The great experimental physicist Michael Faraday made a series of observations that can
be summarized as follows:
If you change the magnetic flux through a loop of wire,
you will induce a current in the wire.
This phenomenon is known as induction. Let’s take a look at some examples:
•
You can try this yourself: http://phet.colorado.edu/en/simulation/faraday
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March 10, 2015
Faraday’s Law of Induction
•
Faraday’s quantitative results can be summarized in a simple law known as his law of
induction:
ε induced = −
ΔΦ B
Δt
What is this equation telling us?
•
What does it mean to talk about the EMF around a loop? Does this “violate” any laws
that you had learned before?
•
So the important idea is changing the flux. How can we change the magnetic flux
through a loop of wire?
 
Φ B = B ⋅ A = BA cosθ
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magnetic flux
Physics E-1ax
time
March
10, 2015
Activity 5: Using Faraday’s Law
I’m going to do a demonstration where I rotate a bar magnet next to a loop of wire and
measure the current in the wire. Sketch below the current observed as a function of time.
current
1.
Now construct a model: how does the flux change in the loop? How does the changing
flux result in a current?
N
S •
N
N
magnetic flux
•
S
S
•
N
N • S
•
S
time
current
2.
time
time
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Physics E-1ax
March 10, 2015
Activity 6: Getting the Sign Right: Lenz’s Law
•
There’s an important minus sign in Faraday’s Law:
ε induced = −
ΔΦ B
Δt
This means that the direction of the induced EMF is “opposite” to the direction of the
change in flux. This sense of “opposite” is made precise in Lenz’s Law:
The induced current flows in whichever direction is needed to
counteract the change in flux of the original magnetic field.
Let’s see what this means with an example:
•
Suppose we move the north pole of a magnet towards a loop, as shown below. What will
be the direction of the induced current?
•
Here’s a general strategy for Lenz’s Law:
1. Choose your area vector to point in the same direction as the original field.
2. This way, the flux will be positive. Is the flux increasing or decreasing?
3. Determine the direction of the induced field to counteract the change in flux.
4. Use the right-hand rule to find the direction of the induced current.
1.
Your turn: what direction will the current flow if you pull the S pole away from the loop?
•
If the flux is increasing, go against the flux! If the flux is decreasing, support the flux!
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Physics E-1ax
March 10, 2015
Activity 7: Calculating using Faraday’s Law
•
Now let’s try a quantitative example using Faraday’s Law:
You have a square loop of wire with sides of length d = 5 cm.
The loop is situated at the edge of a uniform magnetic field
B0 = 0.6 T that points into the page, as shown. You pull the
loop completely out of the field at a constant speed during a
time Δt = 0.1 seconds.
1.
Calculate the initial magnetic flux and the final magnetic flux
through the loop. (Hint: what direction will you choose for
the area vector?)
Φinitial =
Φfinal =
2.
Now use Faraday’s Law to find the induced EMF. (Hint: what is ΔΦ Δt ?)
3.
If the loop of wire has a resistance of R = 100 Ω, calculate the magnitude and direction of
the induced current. (You’ll need to use Ohm’s Law and Lenz’s Law here…)
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March 10, 2015
Am I getting it?
1.
The magnetic field shown in the figure at right points into the page, and
is getting weaker. What is the direction of the current induced in the
wire loop?
a) clockwise
b) counterclockwise
c) there is no induced current
2.
The magnet in the figure at right starts off at rest. You then pull the
magnet quickly to the right. What is the direction of the current
induced in the loop?
a)
3.
b)
Following up on question 2 above: Will there be a force on the bar magnet due to the
current induced in the loop? If so, what direction will the force point?
a) There will be no force on the magnet from the loop.
b) The force on the magnet will point to the right.
c) The force on the magnet will point to the left.
4.
In the figure at right, loop 1 has a current flowing clockwise. Loop
2 is aligned on-axis with loop 1, and has no current (the loops are at
rest). What will be the direction of the current induced in loop 2 if
you move the loops closer together?
a) clockwise
b) counterclockwise
c) there is no induced current
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Physics E-1ax
March 10, 2015
One-Minute Paper
Your name: _________________________________
Names of your group members:
_________________________________
_________________________________
•
Please tell us any questions that came up for you today during lecture. Write “nothing”
if no questions(s) came up for you between 6–9pm (or while viewing it online).
•
What single topic left you most confused after today’s class?
•
Any other comments or reflections on today’s class?
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