Ampere

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Amperes Law
2
Outline

• Introduce Amperes Law as an analogy to Gauss
  Law.
• Define Amperes Law.
• Show how to use Amperes Law for cases with
  symmetry.

3
Student Objectives

• Recognise Amperes Law to be analogous to Gauss
  Law.
• Recognise similar concepts (1) draw an imaginary
  shape enclosing the current carrying conductor,
  (2) current enclosed.

4
Amperes Law

• Gauss law allowed us to find the net electric
  field due to any charge distribution (with little
  effort) by applying symmetry.
• Similarly the net magnetic field can be found
  with little effort if there is symmetry using
  Amperes law.

5
Amperes Law

• Amperes law,
• Where the integral is a line integral.
• B.ds is integrated around a closed loop called an
  Amperian loop.
• The current ienc is net current enclosed by the
  loop.

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Amperes Law

• ie,
• ie ienc

7
Amperes Law

• The figure shows the cross section of 3 arbitrary
  long straight wires with current as shown.

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Amperes Law

• Two of the currents are enclosed by an Amperian
  loop.

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Amperes Law

• An arbitrary direction for the integration is
  chosen.

10
Amperes Law

• The loop is broken into elements of length ds
  (choose in the direction of the integration).
• Direction of B doesnt need to be known before
  the integration!

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Amperes Law

• B can be in an arbitrary direction at some angle
  to ds as shown (from the right hand grip rule we
  know B must in the plane of page).
• We choose B to be in the
• direction as ds.

12
Amperes Law

• The right hand screw (grip) rule is used to
  assign a direction to the enclosed currents.
• A current passing through the loop in the same
  direction as the thumb are positive ( in the
  opposite direction -ve).

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Amperes Law

• Consider the integral,

14
Amperes Law

• Applying the screw rule,

15
Amperes Law

• Example. Find the magnetic field outside a long
  straight wire with current.

I
r
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Amperes Law

• We draw an Amperian loop and the direction of
  integration.

Direction of Integration
17
Amperes Law

• Recall,
• Therefore,
• The equation derived earlier.

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Amperes Law

• The positive sign for the current collaborates
  that the direction of B was correct.

19
Amperes Law

• Example. Magnetic Field inside a Long Straight
  wire with current.

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Amperes Law

• Amperes Law,

21
Amperes Law

• Amperes Law,
• The charge enclosed is proportional to the area
  encircled by the loop,

22
Amperes Law

• The current enclosed is positive from the right
  hand rule.