Download Magnetic Fields on Current-Carrying Wires Sources of the Magnetic

Magnetic Fields on Current-Carrying Wires
Sources of the Magnetic Field
If a magnet exerts a force on a current carrying wire, doesn’t Newton’s Third Law
apply? Where is the equal and opposite force? If moving charge feels a force it
should exert a force on the source of the magnetic field. Oersted set out to
investigate magnetic phenomena, including this idea.
Hans Christian Oersted
Hans Christian Oersted in giving a lecture at Copenhagen University accidentally
found that the wire across a compass needle caused the compass needle to deflect
when current was flowing. Andre Marie Ampere heard of Oersted’s discovery and
began to experiment on current carrying wire and discovered that the magnetic
force experienced by a compass needle close to a current carrying wire acts at right
angles to the current. The magnetic field around the current carrying wire forms
concentric circles.
The Second Right Hand Rule
Ampere’s discovery led to the second right hand rule. We use this to determine the
direction of the magnetic field around a current carrying wire. The thumb of the
right hand points in the direction of the conventional current and the fingers curl in
the direction of the magnetic field around the wire.
Second Right Hand Rule
Here you see the thumb points in the direction of the current and the fingers curl
around the wire in the direction of the magnetic field.
Magnetic Field around a Straight Current Carrying Wire
A straight current carrying wire produces a circular or more accurately a cylindrical
magnetic field in the space surrounding it. That field is constant at any given
perpendicular distance from the wire and decreases as this distance increases. BiotSavart discovered that the magnetic field is proportional to the current in the wire
and inversely proportional to the distance from the wire.
Magnetic Field Outside a Long Straight Current Carrying Wire
B = µoI/2πr
B is the magnetic field. I is the current. R is the distance from the wire. µo is the
permeability of free space. µo = 4π x 10-7 Tm/A.
Problem
Two long current carrying wires are directed into the page as shown. If wire 1
carries a current of 3.00 Amps and wire 2 carries a current of 3.00 Amps what is
the magnetic field at point P?
Solution
We must first find the magnetic field for each wire. B1 = 5 x 10-6 T. B2 = 1.2 x 105
T.
Now we must break them into components. So here we have found the x and y
components of each field vector at point p. We add the x components together and
the y components together. This gives us our resultant magnitude of 13.0 µT. The
direction is in the –x. Our answer is -13.0 µT i.
Biot-Savart
dB=µo/4π(ids x r/r2) This equation is the Biot-Savart Law. It was derived from
experimentation. We will use it to calculate net magnetic field produced at a point
by various current distributions.
Using Biot-Savart on a Long Straight Current Carrying Wire
As you see we obtain the same result for a long straight current carrying wire B =
µoI/2πR.
Circular Arc of Wire
Here we use Biot-Savart. This gives us B = µoI/4πR integral of theta from 0 to
theta. So B = µoIΘ/4πR.
Ampere’s Law
Remember Gauss’ Law with symmetry we were able to find the electric field with
less difficulty. With Ampere’s Law we can find the magnetic field with less difficulty
as well. Ampere’s Law is derived from Biot-Savart and is credited to Andre Marie
Ampere; however James Clerk Maxwell was the physicist to advance this law.
Ampere’s Law
The integral around a closed loop of B dot ds equals µoI enclosed. Similar to Gauss’
Law, we integrate around a closed loop, called an Amperian Loop. The current
enclosed on the right side of the equation is the net current encircled by the loop.
Right Hand Rule for Ampere’s Law
Curl your right hand around the Amperian loop in the direction of the integration.
The thumb points in the direction of positive current and the opposite direction is
negative current.
Here you see the fingers curled in the direction of the loop and the thumb points up
in the direction of the positive current.
Long Straight Current Carrying Wire
Let’s look at the long straight current carrying wire using Ampere’s Law. Again we
obtain the same result for outside the wire.
Inside the Current Carrying Wire
Here we integrate in a circular path. The current enclosed is I (πr2/πR2). B =
µoIr/2πR2. This is the magnetic field inside the current carrying wire.
Current Carrying Loop
Ampere decided to experiment with bending the wire into a loop. This concentrates
the magnetic field inside the loop so the magnetic field is stronger inside the loop
and weaker outside the loop. Now imagine curling several loops of wire together.
This makes a solenoid.
Solenoids
Solenoids are coils of wire wrapped around some type of core. The core may be air
or it may be a metal. The metal core makes the solenoid a stronger magnet.
Solenoids produce magnetic fields around them when current passes through them.
The magnetic field is stronger within the core if the core is metal instead of air.
Solenoid Uses
Solenoids are electromagnets. A solenoid switch is known as a relay which is an
electrical switch. These are used in automobile starters.
Ampere’s Law applied to Solenoids
Now we need to apply Ampere’s Law to the solenoid. We will need to sketch our
Amperian Loop. Here you see the loop abcd with side length h.
Amperian Loop
Now we have our loop and we take it to be abcda to complete our loop. We need to
sum the integrals of each loop segment.
The integral around the closed loop of B dot ds equals the integral from a to b of B
dot ds plus the integral from b to c of B dot ds etc. The integral from a to b is Bh
because the magnetic field is constant and the h is the side of the loop from a to b.
The bc and da integrals are both zero. The cd integral is zero because this side lies
outside the loop where the magnetic field is zero.
Magnetic Field Inside a Solenoid
The current enclosed in the loop is the following: I enclosed = inh. This we can put
together with the integrals we just found and we get the equations for the magnetic
field inside a solenoid. Where n is the number of turns per unit length.
Bh = µoinh so B = nµoi
Toroids
A toroid may be described as a solenoid bent into a ring shaped like a doughnut.
Our question here is what magnetic field is set up inside the loops of wire that
make up the toroid? The lines of the magnetic field form concentric circles inside
the toroid. We will choose a concentric circle of radius r as our Amperian loop. We
will move through the loop in the clockwise direction.
Applying Ampere’s Law to a Toroid
Ampere’s Law yields the following:
B = Nµoi/2πr
In contrast to the solenoid the magnetic field of the toroid is not constant over the
cross section of the toroid. The magnetic field outside the toroid is zero. The
direction of the magnetic field within the toroid follows from the curled right hand
rule we used for Ampere’s Law.
Problem
A 200 turn solenoid has a length of 0.25 m and a diameter of 0.10 m carries a
current of 0.30 A. Find the magnetic field inside the solenoid.
Solution
B = nµoi
B = 3.0 x 10-4 T