## Current Density

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# Current density

Consider a line charge λ travelling down a wire at speed v. A segment of length vΔt, carrying charge λvΔt, passes by point A in a time interval ∇t. The current in a wire is the charge per unit time passing a given point.I = dq/dt = λv Δt/Δt = λv … (1)

The magnetic force on a segment of current-carrying wire is

Fmag = ∫(λdl)(v x B = ∫ (I x B) dl … (2)

Since I and d I both point in the same direction,

Fmag = ∫ I(d I x B) … (3)

Surface Current Density. If, for some reason, the moving charges can be thought of as being constrained to flow on a surface, we can define a surface current density

**K**. Its direction of flow of charge and its magnitude K is defined as equal to the current per unit length through a line lying in the surface and unit length through a line lying in the surface and oriented perpendicular to the flow. Consider a “ribbon” of infinitesimal width dl |, running parallel to the flow. Let the current in the ribbon be d

**I**.

. : F = dI/dl

__|__

Let the surface charge density be σ and its velocity be v.

The net line charge on the ribbon = σ dl|

dI = (σ dl

__|__)v.

. :

**K**= σ

**v**… (5)

The magnetic force on a surface current is

**F**

_{mag}= ∫(σ da)(

**v**x

**B**) = ∫(

**K**x

**B**)da … (6)

**Volume Current Density (J).**When the flow of charge is distributed throughout a three-dimensial region, we describe it by the volume current density J. Its direction is that of the direction of flow of charge and its magnitude J is given by the current per unit area through an area set perpendicular to the flow. Consider a “tube” of infinitesimal cross section da|, running parallel to the flow. Let the current in this tube be dI. Then

**J**= dI/da| … (7)

Let the volume charge density be ρ,

Let the velocity of the moving charges be v. Then,

**J**= ρ

**v**… (8)

The magnetic force on a volume current is

**F**

_{mag}= ∫(ρ da)(

**v**x

**B**) = ∫(

**J**x

**B**) dτ … (9)

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