Three Chapter 6

Ampere's Law

• The fields discussed so far are uniform fields.
• Fields can vary in
  • Magitude from place to place
  • Direction from place to place
• It is important to be able to determine the field strength anywhere in space given the conditions creating the field.
• Consider a current carrying wire
  • It has a circular magnetic field around it.
  • Direction is determined by the right hand rule.
  • Magnitude of the magnetic field decreases as distance from the wire increases.

• Ampere worked on establishing this relationship
• Ampere's Law - Along any closed path through a magnetic field, the sum of the products of the components of the magnetic field parallel to the path segment with the length of the segment is directly proportional to the net electric current passing through the area enclosed by the path.

• The contents of the summation sign is a dot product. This means that the value of B is the component of the magnetic field parallel to the length Dl. Dl is one of the small segments of the enclosing path. I = current and m_o = 4p X 10^-6 T.m/A. The symbol S means sum of.

• The above equation gives the magnetic field strength (B) at a distance r from a current carrying wire with a current of I flowing through it.
• Now Consider a solenoid of length L with N loops of wire and current I in the loops.

• The closed path we choose is WXYZ which can be broken up into segments WX, XY, YZ, and ZW.
• The total current flowing through the closed path is NI. That is, the current through each loop times the number of loops.
• Along WX and YZ, the magnetic field B is nearly perpendicular to the path therefore

• Along ZW, the magnetic field B, is very small compared to the path YX therefore we can ignore it. I.e.

• Along XY, B (the magnetic field) is straight, constant and parallel to XY. Therefore, B_xy = B
• Applying Ampere's law.

or

m_o NI = B_xyLxy + B_yzL_yz + B_xwL_xw + B_zwL_zw

m_o NI = BL_xy + 0 + 0 + 0

Therefore,

m_o NI = BL or

• This last equation gives the magnetic field strength (B) inside a solenoid of length L, with N loops of wire, and current I flowing through the loops. The direction of B is determined by the right hand rule as discussed earlier.

Ampere as a Unit of Current

• Ampere - is equal to a current of 1 C/s.
  • This comes from the formula

• This idea of an ampere is not practical because Q (Charge) is not easily measured.
• The Ampere is in reality defined in terms of the forces between two parallel current carrying conductors.
• Consider two wires with current I₁ and I₂ in them. The distance between the wires is d.

• In the above equation, if F > 0, then the force is attractive. This understanding comes from applying the right hand rules.
• Generally: if the currents are in the same direction, then the wires are attracted to each other, and if in opposite directions, they will repell each other.
• This equation has been used to define the Ampere.
• Here we have the formally accepted definition of the Ampere.
  • Ampere - The current through each of two long, straight, parallel conductors one meter apart in a vacuum, when the magnetic force between them is 2 X 10^-7 N per meter of length.
• Now that the Ampere has a formal definition which can be used experimentally determine its value, we can re-define the unit of charge (Coulomb)
  • Coulomb - The charge transported by a current of one Ampere in a time of one second.

1 C = 1 A.s

January 16, 2014