{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

8.022pset10solution

# 8.022pset10solution - Massachusetts Institute of Technology...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Massachusetts Institute of Technology Department of Physics 8.022 Spring 2004 Solution 10: Displacement current, Maxwell’s equations, Radiation. TA: Yi Mao, [email protected] 1. Purcell 9.6: Radiation’s ~ E and ~ B in SI units. The source free Maxwell Equations read as follows. ~ ∇× ~ E =- ∂ ~ B ∂t (1) ~ ∇× ~ B = μ ² ∂ ~ E ∂t (2) ~ ∇· ~ E = (3) ~ ∇· ~ B = . (4) Consider the electromagnetic wave of the form ~ E = ˆ zE sin( y- vt ) ~ B = ˆ xB sin( y- vt ) . (5) Plug eqs.(5) into eqs.(1-4). Eqs.(3) and (4) are automatically satisfied. Eqs.(1) and (2) become ˆ xE cos( y- vt ) = ˆ xB v cos( y- vt )- ˆ zB cos( y- vt ) =- ˆ zμ ² E v cos( y- vt ) or, E = B v (6) B = μ ² E v. (7) From Eqs.(6) and (7) one can derive the value of speed v of electromagnetic wave in the vacuum: v = 1 / √ ² μ . (8) 2. Purcell 9.8: Wave in a box. For the EM field ~ E = ˆ zE cos kx cos ky cos ωt (9) ~ B = B (ˆ x cos kx sin ky- ˆ y sin kx cos ky ) sin ωt, (10) the two divergencelessness equations of Purcell eqs.(16) ~ ∇· ~ E = 0 ; ~ ∇· ~ B = 0 , can be easily verified. The other two equations give ~ ∇× ~ E + 1 c ∂ ~ B ∂t = ( ωB c- E k )(ˆ x cos kx sin ky- ˆ y sin kx cos ky ) cos ωt = 0 ~ ∇× ~ B- 1 c ∂ ~ E ∂t = ( E ω c- 2 B k )ˆ z cos kx cos ky sin ωt = 0 or E = ω kc B (11) E = 2 kc ω B . (12) So the condition is that ω = √ 2 ck (13) E = √ 2 B . (14) The magnetic field inside a square metal box of size π/k is shown in Figure 1. Note that the box must run from- π/ (2 k ) to π/ (2 k ) in both x and y in order to satisfy the...
View Full Document

{[ snackBarMessage ]}

### Page1 / 6

8.022pset10solution - Massachusetts Institute of Technology...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online