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 A changing magnetic field induces an EMF A changing electric flux induces magnetic field ! A changing magnetic flux induces an electric field !

13. Displacement current 1) Ideas (Maxwell) Faraday law: Changing magnetic field (more exactly, flux) produces electric field Maxwell’s idea: Changing electric field (more exactly, flux) produces magnetic field 2) Problem The current I ≠ 0 throughout the flat surface below, but I = 0 throughout the curved surface. This is in contradiction with the Ampere’ law: Q(t) Ic(t) 3) Solution Alternating current can flow in a circuit with a capacitor

Example: A circular parallel-plate capacitor with plates 2 Example: A circular parallel-plate capacitor with plates 2.0cm in diameter is accumulating charge at the rate of 3.50 mC/s at some instant of time. What is the magnitude of the induced magnetic field at the distance r measured radially outward from the center of the plates? a) r=10.0 cm; b) r=1.0 cm Q(t) r R Ic(t) rb a) b)

14. Electromagnetic waves Maxwell’s equations and electromagnetic waves Properties of electromagnetic waves Speed of light: Wave length: Energy density: for e.m. wave for any e.m. field (U – energy, V - volume) Intensity:

(Polarizeation of transverse waves) 15. Polarization (Polarizeation of transverse waves) 1) Waves on string (polarization and polrizing filters)

2) Electromagnetic waves (polarized light) y This wave is polarized in y direction x direction of motion of wave z e.m. waves are transverse waves Light is polarized when its electric fields oscillate in a single plane, rather than in any direction perpendicular to the direction of propagation. 3) Unpolarized light Unpolarized light consist of waves with randomly directed electric fields. Here the waves are all traveling along the same axis, directly out of the page, and all have the same amplitude E. A second way of representing unpolarized light – the light is the superposition of two polarized waves whose planes of oscillation are perpendicular to each other. - intensity of unpolarized light I - intensity of polarized component

4) Polarisation of light (Malus’s law) When light passes through a polarizer, only the component parallel to the polarization axis is transmitted. If the incoming light is plane-polarized, the outgoing intensity is: Example (two sheets): The light transmitted by polarizing sheet P1 is vertically polarized, as represented by the vertical double arrow. The amount of that light that is transmitted by polarizing sheet P2 depends an the angle between the polarization direction of that light and the polarizing direction of P2

Polarized light will not be transmitted through a polarized film whose axis is perpendicular to the polarization direction. This means that if initially unpolarized light passes through crossed polarizers, no light will get through the second one. Example (three sheets):

Classification of Electromagnetic Waves Wavelength decreases  Frequency increases  Note: 1 nanometer = 10-9 meter

Example1: An electromagnetic wave in vacuum has a frequency of 1500 KHz. What is the wavelength of the wave? Example2: An electromagnetic wave in vacuum is moving in +y direction. At time t=0 and at position (x,y,z)=(0,0,0), the electric field is pointing in the +z direction. In what direction is pointing the magnetic field at that time and position? y x z