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Maxwell’s Equations

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

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Maxwell’s Equations

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  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 e = N(DF/Dt) Vp/Vs = Np/Ns • Where p and s are the primary and secondary solenoids

  4. Transformers and Current • If Np > Ns, voltage decreases (is stepped down) • Energy is conserved in a transformer so: • IpVp = IsVs • Decrease V, increase I

  5. Transformer Applications • Voltage is stepped up for transmission • Since P = I2R 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 = -dFB/dt • ∫ B ds = m0e0(dFE/dt) + m0ienc • ∫ E dA = qenc/e0 • Gauss’s Law for Magnetism • ∫ B dA = 0

  7. Faraday’s Law • ∫ E ds = -dFB/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 = m0e0(dFE/dt) + m0ienc • The second term (m0ienc) is Ampere’s law • The first term (m0e0(dFE/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, id id = e0(dFE/dt) ∫ B ds = m0id + m0ienc

  10. Displacement Current in Capacitor • So then dFE/dt = A dE/dt or id = e0A(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 = qenc/e0 • 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 w to increase the rms current through a RC circuit? • Increase all three • Increase R and C, decrease w • Decrease R, increase C and w • Decrease R and w, increase C • Decrease all three

  16. How would you change R, L and w to increase the rms current through a RL circuit? • Increase all three • Increase R and L, decrease w • Decrease R, increase L and w • Decrease R and w, increase L • Decrease all three

  17. How would you change R, L, C and w to increase the rms current through a RLC circuit? • Increase all four • Decrease w and C, increase R and L • Decrease R and L, increase C and w • Decrease R and w, increase L and C • None of the above would always increase current

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