1. Maxwell’s Equations PH 203 Professor Lee Carkner Lecture 25

2. Transforming Voltage   We often only have a single source of emf   We need a device to transform the voltage   Note that the flux must be changing, and thus the current must be changing  Transformers only work for AC current

3. Basic Transformer   The emf then only depends on the number of turns in each  = N(  /  t)  V p /V s = N p /N s  Where p and s are the primary and secondary solenoids

4. Transformers and Current   If N p > N s, voltage decreases (is stepped down)   Energy is conserved in a transformer so:  I p V p = I s V s   Decrease V, increase I

5. Transformer Applications   Voltage is stepped up for transmission  Since P = I 2 R a small current is best for transmission wires   Power pole transformers step the voltage down for household use to 120 or 240 V 

6. Maxwell’s Equations  In 1864 James Clerk Maxwell presented to the Royal Society a series of equations that unified electricity and magnetism and light   ∫ E ds = -d  B /dt   ∫ B ds =  0  0 (d  E /dt) +  0 i enc   ∫ E dA = q enc /  0  Gauss’s Law for Magnetism  ∫ B dA = 0

7. Faraday’s Law   ∫ E ds = -d  B /dt   A changing magnetic field induces a current  Note that for a uniform E over a uniform path, ∫ E ds = Es

8. Ampere-Maxwell Law   ∫ B ds =  0  0 (d  E /dt) +  0 i enc  The second term (  0 i enc ) is Ampere’s law   The first term (  0  0 (d  E /dt)) is Maxwell’s Law of Induction   So the total law means  Magnetic fields are produced by changing electric flux or currents

9. Displacement Current  We can think of the changing flux term as being like a “virtual current”, called the displacement current, i d i d =  0 (d  E /dt)  ∫ B ds =  0 i d +  0 i enc

10. Displacement Current in Capacitor   So then d  E /dt = A dE/dt or i d =  0 A(dE/dt)  which is equal to the real current charging the capacitor

11. Displacement Current and RHR   We can also use the direction of the displacement current and the right hand rule to get the direction of the magnetic field  Circular around the capacitor axis  Same as the charging current

12. Gauss’s Law for Electricity   ∫ E dA = q enc /  0   The amount of electric force depends on the amount and sign of the charge  Note that for a uniform E over a uniform area, ∫ E dA = EA

13. Gauss’s Law for Magnetism   ∫ B dA = 0  The magnetic flux through a surface is always zero  Since magnetic fields are always dipolar 

14. Next Time  Read 32.6-32.11  Problems: Ch 32, P: 32, 37, 44

15. How would you change R, C and  to increase the rms current through a RC circuit? A)Increase all three B)Increase R and C, decrease  C)Decrease R, increase C and  D)Decrease R and , increase C E)Decrease all three

16. How would you change R, L and  to increase the rms current through a RL circuit? A)Increase all three B)Increase R and L, decrease  C)Decrease R, increase L and  D)Decrease R and , increase L E)Decrease all three

17. How would you change R, L, C and  to increase the rms current through a RLC circuit? A)Increase all four B)Decrease  and C, increase R and L C)Decrease R and L, increase C and  D)Decrease R and , increase L and C E)None of the above would always increase current