PROBLEMS 24-1 Maxwell’s Equations1. Gauss ’ laws for electric fields and for magnetic fields differ due to the lack of magnetic charges. Assume that magnetic monopoles (magnetic charges) exist; denote them by the symbol M . Rewrite Gauss ’ law for magnetic fields, and give the SI units for M .2. Equations 24-1 through 24-4 apply in a vacuum. Write Maxwell ’s equations for matter instead by using the dielectric constant εr and the relative permeability μr .24-3 Properties of Electromagnetic Waves3. A plane electromagnetic wave has a maximum electric field of 3.20⨯10-4 V/m. Find the maximum magnetic field.4. If the electric field for a plane electromagnetic wave is given by E x = 0, E y = E 0 cos (kz +ωt ), and E z = 0, what are B and the direction of propagation of the wave?5. The electric field of a certain plane electromagnetic wave is given by E x = 0, E y = 0, and .cos[()],15z E 2010t-x/c π= with c = 3.0⨯108 m/s and all quantities in SI units. Write expressions for the components of the magnetic field of the wave.6. A plane electromagnetic wave of wavelength 1.2 m propagates in the z -direction. The electric field points in the y -direction and has amplitude of 3.0 V/m. Write an expression for the magnetic field, including its amplitude in SI units. Assume that the electric field is at its maximum at z = 0, t = 0.24-4 Energy Transport and the Poynting V ector7. A radio station emits a signal with a power of 50 kW. What are the amplitudes of the electric field and magnetic field at distances of 10 km and 1000 km? Assume that the signal far from the antenna is transmitted with equal intensity in all directions. (Real radio stations cannot afford to transmit their energy in this way, and their antennas distribute energy with a high degree of directionality.)8. The electric field for a given electromagnetic wave has a peak value 50 mV/m. What is the intensity of the wave?9. Assume that a 100 W light bulb emits light equally in all directions. What are the peak values of the electric and magnetic fields at a distance of 1 m?10. A laser delivers 103 J of energy in a pulse that lasts 10-9 s. What are the peak electric and magnetic fields for a laser beam of diameter 10-3 m?11. What is the intensity of a plane traveling electromagnetic wave if B m is 1.0⨯10-4 T?12. The maximum electric field at a distance of 10 m from an isotropic point light source is 2.0 V/m. What are (a) the maximum value of the magnetic field and (b) the average intensity of the light there? (c) What is the power of the source?13. An airplane flying at a distance of 10 km from a radio transmitter receives a signal of intensity 10 μW/m 2. Calculate (a) the amplitude of the electric field at the airplane due to this signal, (b) the amplitude of the magnetic field at the airplane, and (c) the total power of the transmitter, assuming the transmitter to radiate uniformly in all directions.24-5 Radiation Pressure14. A plane electromagnetic wave of wavelength 12 μm and an electric field amplitude of 10 V/m impinges on a totally reflecting surface of area 10 cm 2. What is the radiation pressureexerted by the wave?15. High-power lasers are used to compress a plasma (a gas of charged particles) by radiation pressure. A laser generating pulses of radiation of peak power 1.5⨯103 MW is focused onto 1.0 mm 2 of high-electron-density plasma. Find the pressure exerted on the plasma if the plasma reflects all light beams directly back along their path.16. What is the radiation pressure 1.5 m away from a 500 W light bulb? Assume that the surface on which the pressure is exerted faces the bulb and is perfectly absorbing and that the bulb radiates uniformly in all directions.17. Prove, for a plane electromagnetic wave that is normally incident on a plane surface, that radiation pressure on the surface is equal to the energy density in the incident beam. (This relation between pressure and energy density holds no matter what fraction of the incident energy is reflected.)18. A particle in the solar system is under the combined influence of the Sun ’s gravitational attraction and the radiation force due to the Sun ’s rays. Assume that the particle is a sphere of density 1.0⨯103 kg/m 3 and that all incident light is absorbed. (a) Show that, if its radius is less than some critical radius R , the particle will be blown out of the solar system. (b) Calculate the critical radius.Problems1. 0enc B dA M μ⋅=⎰ , A ⋅m.2. enc r 0q E dA εε=⎰ , 0r1B dA μ⋅=⎰ , 00r 0enc r 1d B ds E dA i dt μεεμμ⋅=⋅+⎰⎰ ,r r d 1E ds B dA dt εμ⋅=-⋅⎰⎰ .3. 1.07⨯10-12 T.4. cos(),m x E B kz t cω=+in the - z direction.5.B x = 0, B z = 0,.cos[(/)].95y B 671010t x c π-=-⨯- 6. 88510sin(510).3z B z t ππ-=--⨯ 7. E m = 1.7⨯10-1 V/m, B m = 5.8⨯10-10 T at 10 km;E m = 1.7⨯10-3 V/m, B m = 5.8⨯10-12 T at 1000 km.8. 3.3⨯10-6 W/m 2.9.E m = 77.5 V/m, B m = 2.58⨯10-7 T.10. 3.1⨯1010 V/m, 1.0⨯102T.11. 1.2⨯106 W/m2.12. (a) 6.7 nT; (b) 5.3 mW/m2; (c) 6.7 W.13. (a) 87 mV/m; (b) 0.30 nT; (c) 13 kW.14. 8.8⨯10-8 Pa.15. 1.0⨯107 Pa.16. 5.9⨯10-8 Pa.17. proof problem.18. (b) 508 nm.。