We already know that a charge creates a vector field, an electric field all around it and when another charge is placed in that electric field, it experiences a force. Similarly, a magnet (or a current) produces a vector field, a magnetic field all around it and when another moving charge (or current) is placed in that magnetic field, it too experiences a force. For magnetic poles, the magnetic field lines go from North to South.

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18 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

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AP Physics C

Magnetic forces and fields Notes


We a
lready

know

that a

charge creates a vector field,

an
electric

field
E
,

all around it and
when another charge is placed in that electric field, it experiences a force. Similarly, a
magnet (or a current) pr
oduces a
vector field
,

a
magnetic field
B
,

all around it and when
another moving charge (or current) is placed in that magnetic field, it too experiences a
force.


For magnetic poles, the magnetic field lines go from North to South.






Let’s first consid
er a moving charge in an external B
-
field. The force that the charge
experiences is given by

F
B

= q (v x B)

Where F is the force, q the charge, v the velocity and B is the strength of the B
-
field.
Note that this is a cross product, so it can also be wri
tten as
F
B

= qv
B

sin


where

is the
angle between the velocity and the B
-
field.


This equation also follows the right hand rule, RHR. For positive charges: Your index finger
represents the direction of the velocity, your middle finger
represents

the direction of the
B
-
fie
ld and your thumb is the direction of the force. For a negative charge, use your left
hand or flip the answer (i.e. if the RHR told you the force is up for a positive charge, it
would be down for a negative charge)


B
-
fields








E
-
fields

-
The B
-
field
only acts on a charge if it is
moving in a direction which is
perpendicular to the direction of the B
-
field.

-
The force will alway
s be perpendicular to
both the B
-
field and the velocity.

-
This force can only change the direction
of motion of the particle;

it cannot speed
it up or slow it down.

-
The E
-
field acts on the charge
regardless of the movement of the
charge.

-
The force is always parallel or anti
-
parallel to the direction of the E
-
field.

-
This force can speed the particle up or
slow it down.


The

unit for B
-
fields is the Tesla = N/(Am)

1 G (Gauss) = 10
-
4

T


Now let’s try some examples of using the RHR.

Since B
-
fields can

affect moving charges, they should also be able to affect current
carrying wires.


We know that the force on a charged particle

is given
by F
B

= q (v x B).

Given a wire of
length l and current I, we can
derive

that the force on a current carrying wire is

F
B

= I (l

x B)


Sometimes, we will want to find the force on a wire where the direction of l will not be
constant, so we’ll hav
e to employ a bit of calculus to the problem. In these cases, you need
to look at the geometry of the wire to see if
any components of the force cancel out

(much
like we did with electric fields)
.

When I isn’t constant, you need to use:




Finding B
-
fields from Currents


One of the most important aspects of this unit is that current carrying wires can produce
magnetic fields! We need to be careful here to distinguish between the magnetic field
produced by a wire and the force
on
a w
ire in an external B
-
field. A current carrying wire
will induce a B
-
field, but it will not feel a force due to its own B
-
field. Another wire (or
charge) in the induced B
-
field, however, will feel a force.


For a
long, straight wire

with current I, the B
-
field induced by the wire at a given point, r,
away from the wire is given by





x 10
-
7
N/A
2

(or Tm/A)


To solve the direction of the
B
-
field
,
simply

put your thumb in the direction of the current
and your fingers

will curl in the direction of the B
-
field.
Note that the B
-
field will wrap
around the wire.








Another commonly used wire is a long wire wrapped across a hollow tube, called a
solenoid.

For an ideal solenoid, where it is much longer than it is wide,

the B
-
field will inside the
solenoid, nearly uniform and
parallel

to the axis. There will be no B
-
field radially outside of
the solenoid. The B
-
field inside the solenoid is given by

B =


nI

Where I is the current in the wire and n is the number of turns/length. Commonly, N is
used for the total number of turns and n is used for the turn density.

To find the direction of the B
-
field, you can do either of two RHRs. The first is the same

RHR as before, where your thumb is in the direction of I, and your fingers point in the
direction of B (just remember that for a solenoid, the B
-
field doesn’t curl, it’s parallel to
the axis). Or, you can curl your fingers in the direction of current and

your thumb points in
the direction of B.








Now, these above equations are for specific cases. A more general form, the
Biot
-
Savart
Law

will help you determine B fields for any shape.




From this equation, you can solve B
-
fie
lds at some point a distance r away from the wire for
variously shaped wires. Again, you will want to look at the geometry of the object

to see
what components of B cancel. You will also need to see if r or the angle between dl and r
changes
because you m
ight want to s
ubstitute in r = (x
2

+ y
2
)
1/2

when appropriate or
substitute in dl = rd

when appropriate.


Ampere’s Law


A
simpler

way to find a B
-
field from current is using ampere’s law. It is very similar to
Gauss’ law in that we can use it when the
situation

involved is symmetric.


Ampere’s Law







Gauss’ Law

To find B fields






To find E fields

.

dl =


I
enclosed







.

dA =

-

Take the integral along a closed
Amperian loop (dl)

-
The loop is always tangent to the B
-
field

-

Take the integral along a closed
Gaussian surface (dA)

-
The surface is always perpendicular to
the E
-
field




Note that in Gauss’ law, we add the charges enclosed if th
ey are positive and subtract them
if they are

negative charges. Similarly, we add the currents in Ampere’s law if they
are
going in the same direction and subtract them if they are going in different directions.




W
e will often need to use current density to find the current enclosed by our Amperian
loop.



units (A/m
2
) or rewritten
,


Like electric flux, there is also an equivalent magnetic flux.


Electric flux =




Magnetic flux =




We will do more with magnetic flux later, but you should know that for a closed surfa
ce
, the
magnetic flux equals zero. This means there are no magnetic monopoles. (unlike electric
fields, where you can just have a positive charge without a negative charge).


In mathematical form, this is written as:



.

dA =
0