Force on a Charge

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Force on a Charge

• Last time we looked at the force exerted on a
  current-carrying wire by a magnetic field
• We found the direction of the force and the
  magnitude of the force
• Now we want to dig deeper and look at the force
  on a moving charged particle

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Force on a Charge

• If N charges of size q pass a point in space in a
  time t they constitute a current INq/t
• If the charge q travels a distance l in time t
  then lvt where v is the velocity of the charged
  particle

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Force on a Charge

• If these charges move in a magnetic field B, then
  FIlBsin?
• Now we substitute lvt, and INq/t
• Thus, Fvt(Nq/t)Bsin?
• So, the force on one charge is qvBsin?
• Of course, ? is the angle between v and B
• Again, use the right hand rule to get directions

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Force on a Charge
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Force on a Charge
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Force on a Charge
When the velocity is not perpendicular to the
field, the path is a spiral. We must resolve the
velocity vector into perpendicular and parallel
components.
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Aurora Borealis
As the particle approaches the magnetic pole, the
field gets stronger and the spiral gets tighter.
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Current in a Straight Wire
The strength of the magnetic field produced by a
current-carrying straight wire depends on the
size of the current. It is also inversely
proportional to the distance from the wire.
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Force Between Parallel Wires

• Since we can think of currents as a collection of
  moving charges, and we know that a
  current-carrying wire creates a magnetic field,
  if we put two current-carrying wires next to each
  other we expect an interaction

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Force Between Parallel Wires

• The current in one wire produces a magnetic field
  that exerts forces on the moving charges in the
  second wire
• The converse is also true
• Lets look at a picture

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Force Between Parallel Wires
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Force Between Parallel Wires
If the currents are parallel, the force is
attractive. If the currents are anti-parallel,
the force is repulsive.
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Definitions

• We finally get around to precisely using the
  modern definition of current and charge
• If I1 I2 1 Ampere and the two wires are 1
  meter apart
• This precisely defines the ampere
• One Coulomb is then one ampere-second

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Amperes Law

• The relationshipis valid only for long straight
  wires.
• We want to find a general relationship in a wire
  of any length and of any shape
• We are going to propose the relationship and then
  demonstrate that it produces the result for a
  long straight wire

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Amperes Law
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Amperes Law
Lets find the magnitude of B at point A which
is a distance r from the wire. We choose a
circular path composed of little arcs around a
circle of radius r. If we choose an infinite
number of little arcs, we can treat them as
little straight lines. Then Amperes Law says
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Amperes Law
VOILA!!! It works and gives exactly what we
expect!
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Amperes Law

• This proposal has been checked for lots of other
  cases, and it always works
• It is regarded as one of the fundamental laws of
  electricity and magnetism
• Shows why we put the 2? into our definition of
  current in the first place

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Amperes Law

• Lets try this law out for another situation
• Well pick a solenoid, a long wire wrapped into
  a coil

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Amperes Law

• Each loop in the solenoid produces a magnetic
  field as we saw before

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Amperes Law
The field inside the solenoid is just the sum of
all the fields produced by the individual loops
of wire. If we pack the coils tightly together,
then the field is parallel to the axis except at
the ends. Now, we just have to set up a closed
path to do the calculation.
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Amperes Law
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Amperes Law
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Amperes Law