Two questions: (1) How to find the force, F on the electric charge, q excreted by the field E and/or B? (2) How fields E and/or B can be created? Maxwell’s equations Gauss’s law for electric field Electric charges create electric field: Gauss’s law for magnetic field Magnetic charges do not exist: For one not moving (v<<c) charge: Amperes law Electric current creates magnetic field: Faraday’s law (Will be discussed later) (As we will see later, this law should be extended)

7. Ampere’s law. (Sources of magnetic field) Iout Iin Iin Iout 2) Applications of Ampere’s law (AL can be used to calculate the magnetic field in situations with a high degree of symmetry) a) Long straight conductor I r Example1: I = 10A r = 0.02 m B - ?

Example2: Two long parallel wires are 4. 0 cm apart Example2: Two long parallel wires are 4.0 cm apart. Each wire carries the current 10 A in the same direction. Find the magnetic field halfway between the wires. r I Example3: Two long parallel wires are 4.0 cm apart. Each wire carries the current 10 A in the opposite direction. Find the magnetic field halfway between the wires. I I r r

Example4: Two long thin parallel wires 13 Example4: Two long thin parallel wires 13.0 cm apart carry 25-A currents in the same direction. Determine the magnitude of the magnetic field at point P, 12.0 cm from one wire and 5.0 cm from the other. The field from each wire makes concentric circles about the source wire. The individual fields at point P are perpendicular to the lines from the wires to point P. Using Pythagorean theorem we have:

Example5: Two long parallel wires, a distance d apart, carry equal currents I in the same direction. One wire is at x=0, the other is at x=d. Determine magnetic field along the x axis between the wires as a function of x. The left wire will cause a field on the x-axis that points in the y-direction, and the right wire will cause a field on the x-axis that points in the negative y-direction. The distance from the left wire to a point on the x-axis is x, and the distance from the right wire is d-x.

b) Long cylindrical conductor (Assuming the current inside the conductor is uniform) R r B r R

N – number of loops (or turns) c) Field inside a solenoid B N – number of loops (or turns) encircled by our path l Example 1: Example 2: The magnetic field inside an air filled solenoid is B. The area of the solenoid is doubled, keeping the current flowing through the solenoid and the number of turns per unit length unchanged. Find the magnetic field inside the new solenoid. Answer: Magnetic field inside a solenoid is independent from the area!

Example 3: A long solenoid has 100 turns/cm and carries current I Example 3: A long solenoid has 100 turns/cm and carries current I. An electron moves within the solenoid in a circle of radius 2.30 cm perpendicular to the solenoid axis. The speed of the electron is 1.40*107 m/s. Find the current in the solenoid.

Magnetic field lines always form a closed loops! Long straight conductor Inside long cylindrical conductor (r<R) Inside solenoid Inside toroidal solenoid (toroid) Magnetic field lines always form a closed loops!

3) Force between two parallel wires 4) Definition of the Ampere The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 newton per meter of length.