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(a) Calculate the (time-averaged) energy density of an electromagnetic plane wave in a conducting medium (Eq. 9.138). Show that the magnetic contribution always dominates.

(b) Show that the intensity is(k2μω)E02e-2xz

Short Answer

Expert verified

(a) The energy density of an electromagnetic plane wave in a conductive medium isu=k22μω2E02e2kz .

(b) It is proved that the intensity is (k2μω)E02e-2xz.

Step by step solution

01

Expression for the energy density of an electromagnetic plane wave:

Write the expression for the energy density of an electromagnetic plane wave.

u=12(εE2+1μB2) …… (1)

Here,ε is the permittivity of free space, E is the electric field, μIs the permeability of free space, and B is the magnetic field.

02

Determine the energy density of an electromagnetic plane wave in a conductive medium:

(a)

Write the expression for the electric field.

E~(z,t)=E0ekzcos(kzωt+δE)x^

Squaring on both sides.

E2=E02e2kzcos2(kzωt+δE)

Write the expression for the magnetic field.

B~(z,t)=B0ekzcos(kzωt+δE+ϕ)y^

Squaring on both sides.

B2=B02e2kzcos2(kzωt+δE+ϕ)

SubstituteE2=E02e2kzcos2(kzωt+δE) andB2=B02e2kzcos2(kzωt+δE+ϕ) in equation (1).

u=12(εE02e2kzcos2(kzωt+δE)+1μB02e2kzcos2(kzωt+δE+ϕ))u=12e2kz[εE02cos2(kzωt+δE)+B02μcos2(kzωt+δE+ϕ)]u=12e2kz[ε2E02+12μB02]

Using the relation between B0andE0 , it is known that:

B0=E0εμ1+(σεω)2

Calculate the energy density of an electromagnetic plane wave in a conductive medium.

u=12e2kz[ε2E02+12μ(E0εμ1+(σεω)2)2]u=14e2kzεE022εμk2ω2u=k22μω2E02e2kz

03

Show that the magnetic contribution always dominates:

Write the expression for the magnetic energy.

umag=(B022μ)12e2kz

Write the expression for the density electrostatic energy density.

uelec=12e2kzε2E02

Take the ratio of magnetic energy density and electrostatic energy density.

umaguelec=(B022μ)12e2kz12e2kzε2E02umaguelec=(B02μ)εE02umaguelec=1εμεμ1+(σεω)2

On further solving,

umaguelec=1+(σεω)2>1umag>uelec

Hence, the magnetic contribution is always the dominator.

Therefore, the energy density of an electromagnetic plane wave in a conductive medium isu=k22μω2E02e2kz .

04

Show that the intensity is (k2μω)E02e-2xz(k2μω)E02e-2xz:

(b)

Write the expression for intensity.

I=cu

Substituteu=k22μω2E02e2kz and c=ωkin the above expression.

I=k22μω2E02e2kzωkI=(ωk)k22μω2E02e2kzI=k2μωE02e2kz

Therefore, it is proved that I=k2μωE02e2kz.

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Suppose string 2 is embedded in a viscous medium (such as molasses), which imposes a drag force that is proportional to its (transverse) speed:

Fdrag=-Yftz.

(a) Derive the modified wave equation describing the motion of the string.

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