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Maxwell's four equations are given as follows:
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?.D = ? ...(1)
?.B = 0 ...(2)
? × E = - ?B/?t ...(3)
? × H = J + ?D/?t ...(4)
However, D = e0E + P and B = µ0(H + M) where D = e0E + P
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J = sE
P = ?e0E
M = ?H
For Vacuum, we have ? = 0, J = 0
Maxwell's four equations simplify to the following form:
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?.E = 0 ...(5)
?.B = 0 ...(6)
? × E = - ?B/?t ...(7)
? × B = (1/c²) ?E/?t ...(8)
Maxwell's differential form in terms of electric and magnetic fields can be written as follows:
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E = E0 e^(i(?t - k.r)) ...(9)
B = B0 e^(i(?t - k.r)) ...(10)
Where,
r = xi + yj + zk = position vector
k = kx i + ky j + kz k
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These (E, B) must simultaneously obey equations (5) to (8).
From equation (5), we can write:
kxEx + kyEy + kzEz = 0 ...(11)
Similarly, from equation (6), we can write:
kxBx + kyBy + kzBz = 0 ...(12)
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Thus, electric field E ? k (direction of propagation)
Similarly, we can show that B can satisfy equation (6) only if
B ? k (electromagnetic wave)
To find relative orientation between E, B, k, we must validate equations (7) and (8) by the relations (9) and (10) of wave equation.
Now we can write:
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? × E = (?Ez/?y - ?Ey/?z)i + (?Ex/?z - ?Ez/?x)j + (?Ey/?x - ?Ex/?y)k
Now (? × E)x = ?Ez/?y - ?Ey/?z
From equation (9), we have:
?Ez/?y = i ky Ez
?Ey/?z = i kz Ey
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(? × E)x = i(kyEz - kzEy)
Similarly, we can find:
(? × E)y = i(kzEx - kxEz)
(? × E)z = i(kxEy - kyEx)
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Put values from equations (7) and (13), we get:
i(kxEy - kyEx) = - ?B/?t
Again from (10), we can show that:
?B/?t = i?B
Put values from equations (14) and (15) in (7), we get:
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i(k × E) = -i?B
or k × E = ?B
Equation (16) shows that B ? k and E.
From equations (5) and (6), it is clear that E, B, k are mutually perpendicular to each other.
Now, we can also satisfy equation (8) as follows:
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? × B = µ0J + µe0 ?E/?t
However, at low temperature of a polarizable dielectric medium:
J = -sE
We can see that:
? × B = i(k × B)
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?E/?t = i?E
Put these values in (8), we get:
i(k × B) = µ0(-sE) + µe0(i?E)
The equation above satisfies Maxwell's equation.
Hence electromagnetic
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This download link is referred from the post: MBBS 1st Year Physiology Most Important Questions From Last 10 Years (Common to all Universities)
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