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 that obeys Ohm's law with a resistivity of $\rho=$ $4.00 \times 10^{-8} \Omega \cdot \mathrm{m} .$ Assume that a potential of $4.00 \mathrm{V}$ is maintained at $x=0,$ and that $V=0$ at $x=0.500 \mathrm{m} .$ Find (a) the electric ficld $\mathbf{E}$ in the wire, (b) the resistance of the wire, (c) the electric current in the wire, and (d) the current density $J$ in the wire. Express vectors in vector notation. (e) Show that $\mathbf{E}=\rho \mathbf{J}$.

Key Concepts

Current and Current Density

Electric current is the rate at which charge flows through a conductor, while current density is the current per unit cross-sectional area of that conductor. Current density provides a more localized measure of how the current is distributed within the material, linking microscopic conduction properties to the macroscopic flow of current.

Electric Field

The electric field represents the force per unit charge exerted on charged particles and is a vector quantity with both magnitude and direction. In the context of a conductor with an applied voltage difference, the electric field is typically uniform along the conductor and drives the drift of electrons.

Vector Notation

Vector notation is used to express quantities that have both magnitude and direction, such as the electric field. In problems involving fields within conductors, expressing these quantities in vector form simplifies the analysis of directional properties along axes or in space.

Resistivity

Resistivity is a material-specific property that quantifies how strongly a material opposes the flow of electric current. It is an intrinsic property independent of the shape and size of the material and plays a key role in determining the resistance of a conductor when combined with its geometry.

Ohm's Law

Ohm's Law is a fundamental principle that relates the electric field and the current density in a material through the material's resistivity. It states that the electric field is proportional to the current density, with the proportionality constant being the resistivity, thereby linking the macroscopic voltage and current to microscopic charge carrier behavior.

Resistance

Resistance is the measure of an object’s opposition to the flow of electric current. It depends on both the material's resistivity and the physical dimensions of the conductor, such as its length and cross-sectional area. It is used in conjunction with Ohm's Law to calculate the current resulting from an applied voltage.

Transcript

00:01 In this question, we have a straight cylindrical wire along the x -axis.

00:07 And then it has a length of 0 .5 meters.

00:15 The diameter is 0 .1 m m.

00:18 And then the resistivity is 4 times the diameter is 0 .2 m.

00:27 Sorry.

00:30 And then the resistivity is 4 times 10 to the power of negative 8 omel meters.

00:34 And then, so this is measured from x equals to the zero, from x equals to l, okay, where l is 0 .5 meters.

00:50 And then here the voltage is 4 volts, and the voltage here is 0 volts.

01:00 Okay, so we want to find the electric field in the wire, resistance, electric current in the wire, current density, and show that e equals to resistivity times the current density.

01:15 So, and then if, and we also need to express vectors in vector notation.

01:22 So in part a, you want to calculate the electric field.

01:26 So using e equals to negative delta v over delta x, x, x, okay? and we have e to be negative, so 0 minus 4 divide by 0 .5 minus 0 xxx and you calculate this to be 8 .0 volt per meter x hat.

01:57 Okay, so this is the answer for part a.

02:03 In part b you want to calculate resistance.

02:07 You'll be using r equals to row l over a.

02:15 Okay.

02:16 And the reservoir the resistance is row l over pi r square...

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