**Convection E****quation Derivation****:**

When the e beam passes through the electric current, it induces a current. This current is known as Convection Current.

The electron velocity, charge density, current density and axial electric field are given by the equations.

v = v

ρ = ρ_{0}+ v_{1}e^{(j}^{ω}^{t-}^{γ}^{z)}^{ }------------------- (a)_{0}+ ρ

_{1 }e

^{(j}

^{ω}

^{t-}

^{γ}

^{z)}

^{ }------------------- (b)

J = -J

_{0}+ J_{1}e^{(j}^{ωt-γz)}^{ }------------------- (c)E

_{γ}= -E_{1}e^{(j}^{ωt-γz)}^{ }-------------------- (d)For a small signal, the current density of electron is given by,

J = ρv

Substitute for ρ and v from eqn (a) and (b)

J = (ρ

_{0}+ ρ_{1 }e^{(j}^{ω}^{t-}^{γ}^{z)})(v_{0}+ v_{1}e^{(j}^{ω}^{t-}^{γ}^{z)})Since the signal is very small, the term e

^{(j}^{ω}^{t-}^{γ}^{z)}is neglected.Therefore, J = (ρ

_{0}+ ρ_{1})(v_{0}+ v_{1})J = ρ

_{0}v_{0}+ ρ_{0}v_{1}+ ρ_{1}v_{0}+ ρ_{1}v_{1}Let ρ

_{1}v_{1 }= 0Therefore, J

_{1}= ρ_{0}v_{0}+ ρ_{0}v_{1}+ ρ_{1}v_{0}Substitute for J from eqn (c), the above equation becomes,

-J

_{0}+ J_{1}e^{(jwt-rz)}= ρ_{0}v_{0}+ ρ_{0}v_{1}+ ρ_{1}v_{0}Let, ρ

_{0}v_{0}= -J_{0}andJ

_{1}= ρ_{0}v_{1}+ ρ_{1}v_{0}---------------------- (e)The axial electric field affects the velocity of electrons as,

dv/dt = -e/mE

_{1}e^{(jwt-rz)}----------- (1)Substitute for v from eqn (a)

dv/dt = ∂/ ∂t(v

_{0}+ v_{1}e^{(j}^{ω}^{t-}^{γ}^{z)}) + ∂z/ ∂t. ∂/∂z(v_{0}+ v_{1}e^{(j}^{ω}^{t-}^{γ}^{z)})dv/dt = v

_{1}e^{(j}^{ω}^{t-}^{γ}^{z)}(jω - γ ∂z/ ∂t)Let, ∂z/ ∂t = v

_{0}Therefore, dv/dt = v

_{1}e^{(j}^{ω}^{t-}^{γ}^{z)}(jω - γ v_{0}) -------------------- (2)Sub eqn (2) in (1)

v

_{1}e^{(jwt-}^{γ}^{z)}(jω - γ v_{0}) = -e/mE_{1}e^{(j}^{ω}^{t-}^{γ}^{z)}so, v

_{1}(jω - γ v_{0}) = -e/mE_{1}v

_{1}= -E_{1}e/m (jω - γ v_{0}) -------------------- (3)eqn (3) defines the change in electron velocity with respect to electric field, E

_{1}.By law of conservation of electric charge, the continuity equation can be written as,

∇. J = - ∂ρ/ ∂t ----------------------- (4)

∇ = ∂/ ∂x + ∂/ ∂y + ∂/ ∂z

Consider only the z component

∇ = ∂/ ∂z

∇. J = - ∂J/ ∂z

Substitute for J from eq (c), we get

∇. J = - ∂/ ∂z(-J

_{0}+ J_{1}e^{(j}^{ω}^{t-}^{γ}^{z)})∇. J = - γJ

_{1}e^{(j}^{ω}^{t-}^{γ}^{z)}------------------------- (5)From eq (b)

- ∂ρ/ ∂t = - ∂/ ∂t(ρ

_{0}+ ρ_{1 }e^{(j}^{ω}^{t-}^{γ}^{z)})- ∂ρ/ ∂t = - ρ

_{1}jω e^{(j}^{ω}^{t-}^{γ}^{z)}----------------------- (6)Substitute eq (5) and (6) in eq (4)

∇. J = - ∂ρ/ ∂t

- γJ

_{1}e^{(j}^{ω}^{t-}^{γ}^{z)}= - ρ_{1}jω e^{(j}^{ω}^{t-}^{γ}^{z)}Therefore - γJ

_{1}= - ρ_{1}jωie, ρ

_{1}= γJ_{1}/ jω --------------------------------- (7)eqn (7) defines the change in charge density ρ with respect to current density J.

Consider eqn (e)

J

_{1}= ρ_{0}v_{1}+ ρ_{1}v_{0}Sub. for v

_{1}from eqn (3) and ρ_{1}from eqn (7)J

_{1}= ρ_{0}(-E_{1}e/m (jω - γ v_{0})) + (γJ_{1}/ jω) v_{0}Thus, J

_{1}= -ρ_{0}eE_{1}ω/m(jω - γv_{0})(ω + γv_{0}j)Multiplying Numerator and Denominator by v

_{0}J

_{1}= -ρ_{0}eE_{1}ω v_{0}/m(jω - γv_{0})(ω + γv_{0}j) v_{0}We have,

J

_{0}= -ρ_{0}v_{0}B

_{e}= ω_{0}v_{0}Therefore,

**J**

_{1}= J_{0}**eE**

_{1}B_{e}**/m(**

**j**

**ω**

**-**

**γ**

**v**

_{0}**)(ω + γ**

**v**

_{0}j**)**

Multiplying numerator and denominator with j

J

_{1}= J_{0}eE_{1}B_{e}j/jm(jω - γv_{0})(ω + γv_{0}j)J

_{1}= J_{0}eE_{1}B_{e}j/m(-ω - γv_{0}j)(ω + γv_{0}j)= J

_{0}eE_{1}B_{e}j/m((jω)^{2}- 2ω γv_{0}j + (γv_{0})^{2})= J

_{0}eE_{1}B_{e}j/m(jω - γv_{0})^{2}= J

_{0}eE_{1}B_{e}j/ v_{0}^{2}m(jω/ v_{0}- γ)^{ 2}= J

_{0}eE_{1}B_{e}j/ v_{0}^{2}m(jB_{e}- γ)^{ 2}[Since ω/ v_{0}= B_{e}]The uniform velocity v

_{0}is given by the eqn,v

_{0}= √(2e v_{0}/m)v

_{0}^{2}= 2e v_{0}/me/ m v

_{0}^{2}= 1/2 v_{0}Therefore

**J**_{1}=**J**_{0}**E**_{1}B_{e}j/ 2v_{0}(jB_{e}**- γ****)**^{ 2}When the electric field is uniform, J

_{1}= iHence,

**i =****J**_{0}**E**_{1}B_{e}j/ 2v_{0}(jB_{e}**- γ****)**^{ 2}This is the equation for convection current. This equation is also known as

**‘electronic equation’**as it determines the current induced by the electric field.
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