Induced EMFs and Electric Fields

1. Induced EMFs and Electric Fields AP Physics C Montwood High School R. Casao

2. A changing magnetic flux induced an EMF and a current in a conducting loop. • An electric field is created in the conductor as a result of the changing magnetic flux. • The law of electromagnetic induction shows that an electric field is always generated by a changing magnetic flux, even in free space where no charges are present. • The induced electric field has properties that are very different from those of an electrostatic field produced by stationary charges.

3. Consider a conducting loop of radius r in a uniform magnetic field that is perpendicular to the plane of the loop. • If the magnetic field changes with time, Faraday’s law tells us that an EMF given by is induced in the loop.

4. The induced current produced implies the presence of an induced electric field E, which must be tangent to the loop since all points on the loop are equivalent. • The work done in moving a test charge q once around the loop is equal to W = q·EMF.

5. The electric force on the charge is F = q·E, the work done by this force in moving the charge around the loop is W = q·E·2·π·r, where 2·π·r is the circumference of the loop. • The two equations for work are equal to each other: q·EMF = q·E·2·π·r, so

6. Combining this equation for the electric field, Faraday’s law, and the fact that magnetic flux Φm = B·A = B·π·r2 for a circular loop shows that the induced electric field is: • The negative sign indicates that the induced electric field E opposes the change in the magnetic field.

7. An induced electric field is produced by a changing magnetic field even if there is no conductor present. • A free charge placed in a changing magnetic field will experience an electric field of magnitude: • The EMF for any closed path can be expressed as the line integral of over the path.

8. The electric field E may not be constant, and the path may not be a circle, therefore, Faraday’s law of induction can be written as: • The induced electric field E is a non-conservative, time-varying field that is generated by a changing magnetic field. • The induced electric field E can’t be an electro-static field because if the field were electrostatic, hence conservative, the line integral of over a closed loop would be zero (dΦm/dt = 0).

9. Electric Field Due to a Solenoid • A long solenoid of radius R has n turns per unit length and carries a time-varying current that varies sinusoidally as , where Io is the maximum current and ω is the angular frequency of the current source. • A. Determine the electric field outside the solenoid, a distance r from the axis.

10. Take the path for the line integral to be a circle centered on the solenoid. • By symmetry, the magni-tude of the electric field E is constant and tangent to the loop on every point of radius r. • The magnetic flux through the solenoid of radius R is:

11. Applying Faraday’s law:

12. The electric field E is constant at all points on the loop:

13. The magnetic field inside the solenoid is: B = μo·n·I • Substituting:

14. The electric field varies sinusoidally with time, and its amplitude fall off as 1/r outside the solenoid. • B. What is the electric field inside the solenoid, a distance r from its axis?

15. Inside the solenoid, r < R, the magnetic flux through the integration loop is Φm = B·π·r2.

17. The amplitude of the electric field inside the solenoid increases linearly with r and varies sinusoidally with time.