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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Maxwell's four equations are given as follows:
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∇.D = ρ ...(1)
∇.B = 0 ...(2)
∇ × E = - ∂B/∂t ...(3)
∇ × H = J + ∂D/∂t ...(4)
However, D = ε₀E + P and B = μ₀(H + M) where D = ε₀E + P
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J = σE
P = χₑε₀E
M = χₘH
For Vacuum, we have ρ = 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(ωt - k.r)) ...(9)
B = B₀ e^(i(ωt - 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 = iωB
Put values from equations (14) and (15) in (7), we get:
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i(k × E) = -iωB
or k × E = ωB
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 + με₀ ∂E/∂t
However, at low temperature of a polarizable dielectric medium:
J = -σE
We can see that:
∇ × B = i(k × B)
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∂E/∂t = iωE
Put these values in (8), we get:
i(k × B) = μ₀(-σE) + με₀(iωE)
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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