Volumetric Current Density in Moving Cylinder

What is the formula for calculating the volumetric current density in a short wooden cylinder with non-uniformly distributed charge?

Formula for Volumetric Current Density

The volumetric current density J can be expressed as: J = I/V = (I/L²)R = (Q/RL³)e(N/L³)αr. The volumetric current density J is independent of the angular acceleration α, so it remains constant throughout the motion of the cylinder. The current can be expressed as: I = (Q/L³)e(N/L³)at.

How can the volumetric current density be calculated when the cylinder moves parallel to its axis with uniform acceleration?

Calculation of Volumetric Current Density in Parallel Motion

Part (a): When the cylinder moves parallel to the axis with uniform acceleration a, the current flows due to the motion of charges inside the cylinder. The force acting on the charges is given by F = ma. The current I can be expressed as I = neAv, where n is the number density of charges, e is the charge of each charge carrier, A is the cross-sectional area of the cylinder, and v is the velocity of the charges. The charge Q is non-uniformly distributed in the volume, but squared with the length, so the charge density is given by ρ = Q/L³. The number density of charges is n = ρ/N, where N is Avogadro's number. The volumetric current density J can be expressed as: J = I/V = (I/L²)R = (Q/RL³)e(N/L³)a. The volumetric current density J is independent of the acceleration a, so it remains constant throughout the motion of the cylinder.

How can the volumetric current density be calculated when the cylinder rotates around the axis with uniform angular acceleration?

Calculation of Volumetric Current Density in Rotational Motion

Part (b): When the cylinder rotates around the axis with uniform angular acceleration a, the current flows due to the motion of charges inside the cylinder. The volumetric current density J can be found as: J = I/V, where I is the current that flows through the cross-sectional area of the cylinder and V is the volume of the cylinder.
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