A straight, cylindrical wire lying along the x axis has a...

A straight, cylindrical wire lying along the $x$ axis has a length of 0.500 $\mathrm{m}$ and a diameter of $0.200 \mathrm{mm} .$ It is made of a material described by Ohm's law with a resistivity of $\rho=4.00 \times 10^{-8} \Omega \cdot \mathrm{m} .$ Assume a potential of 4.00 $\mathrm{V}$ is maintained at the left end of the wire at $x=0 .$ Also assume $V=0$ at $x=0.500 \mathrm{m}$ . Find $(a)$ the magnitude and direction of the electric field in the wire, (b) the resistance of the wire, (c) the magnitude and direction of the electric current in the wire, and (d) the current density in the wire. (e) Show that $E=\rho J$.

Key Concepts

Electric Field and Potential Gradient

The electric field in a conductor is linked to the spatial rate of change of the electric potential. In a uniformly applied potential difference across a conductor, the field is constant and can be calculated using the potential difference divided by the length of the conductor. This relationship is fundamental in understanding how voltage drives charge movement along the wire.

Resistivity and Resistance

Resistivity is an intrinsic material property that measures how strongly a material opposes the flow of electric current. The resistance of a conductor depends on its resistivity, length, and cross-sectional area, and is given by the formula R = ?L/A. This concept is crucial to determining how much voltage will drop along the conductor and what current it will carry.

Ohm’s Law in Differential Form

Ohm’s law relates the local electric field within a material to the current density, establishing that E = ?J. This microscopic formulation of Ohm’s law provides insight into how the material’s characteristics affect the movement of charge, and it directly connects the concepts of electric field and resistivity to the resulting current distribution.

Current and Current Density

Electric current is the net rate of charge flow through a conductor, while current density is the current per unit cross-sectional area. The relationship I = J * A allows one to determine the total current when the current density and the cross-sectional area of the conductor are known. This concept is key for analyzing how uniformly the current flows through a wire.

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