Current density is one of many characteristics that can be used to describe an electric circuit. On a macroscopic scale, we can determine the current that is flowing through wires: it gives us the bigger picture of the flow of charges as a group. However, the amount of current flowing through a wire is often dependent on the shape and size of the wire. If we want a more general understanding of the current in a circuit, which can be applied to different wire dimensions, its useful to determine the current density. That's where the current density is practical, as it accounts for the exact dimensions of the wire, providing us with useful information about every charge that passes through it. In this article, we'll explain exactly what current density is, and derive various equations used to calculate it.
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What is the magnetic field inside a conductive wire? Here, B is magnetic field, I is the current, \(\mu\) is vacuum permeability, r is radius inside the wire, and R is radius outside the wire.
The direction of the current density vector \(\vec{J}\) is defined as the direction of negatively charged particles in an electric field.
A potential difference in a conductive wire produces a perpendicular electric field.
In what direction do electrons flow in a conductive wire?
What kind of quantity is electric current?
What is the equation for current in a non-uniform electric field?
The electric field in a conductive wire is proportional to the resistivity of the conductor and the current density.
What's the sign of the electron current density?
How many different cases of magnetic field can be distinguished in a conductive wire?
What is the definition of drift velocity?
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Team Current Density Teachers
To explain volume current density, we first must understand electric current.
Electric current is the rate at which charge passes through a cross-sectional area of a wire.
It's a macroscopic quantity that describes the behavior of charge made up of many electrons, rather than considering them individually. Mathematically, it can be expressed as
\[I=\frac{\mathrm{d}q}{\mathrm{d}t},\]
where \(I\) is current and \(q\) is the charge.
However, if we want to know how much current passes through a wire independent of the size of the cross-sectional area, we use current density.
Current density is the flow of charge per unit area.
Current density is often referred to as volume current density. Let's take a look at how a current density arises.
Let's imagine a long cylindrical wire with a cross-sectional area \(A\) pictured in Figure 1 below.
Fig. 1 - Potential difference in a conductive wire produces a parallel electric field.
Creating a potential difference between the two ends of this wire will result in an electric field inside the wire. The direction of the field lines will be parallel to the walls of the cylinder; the same as the direction of the electric current.
Considering their negative charge, electrons will be attracted to the positively charged end of the wire, i.e. moving in the opposite direction.
Let's look at all the relevant equations regarding current density on its own and in relation to the electric field.
In its simplest forms, the current density can be expressed as
\[J=\frac{I}{A}.\]
This equation can be rearranged to obtain the formula for current in a uniform electric field:
\[I=JA,\]
while in a non-uniform electric field, it'll be
\[I= \oint \vec{J}\,\mathrm{d}\vec{A}.\]
The electric field in the wire from Figure 1 is proportional to the resistivity of the conductor and the current density. That means that the equation for current density can be written as
\[\vec{J}=\frac{\vec{E}}{\rho},\]
where \(\vec{J}\) is the current density measured in amperes per meter squared \(\left ( \frac{\mathrm{A}}{\mathrm{m}^2}\right )\), \(\vec{E}\) is the electric field with the units of volts per meter \(\left ( \frac{\mathrm{V}}{\mathrm{m}}\right )\), and \(\rho\) is the resistivity measured in Ohm-meters \(\left ( \frac{\mathrm{\Omega}}{\mathrm{m}}\right )\).
The direction of the current density vector \(\vec{J}\) is defined as the direction of positively charged particles in an electric field. That means that electrons have a negative current density: \(\vec{-J}\).
Finally,
\[\vec{J}=nq\vec{v}_\mathrm{d}\]
describes current density using the number of charge carriers \(n\) and drift velocity \(v_\mathrm{d}\). This shows the microscopic nature of current density.
Drift velocity is the average velocity of charged particles in a material due to its electric field.
First, let's define a magnetic field.
Magnetic field is a vector field that describes the magnetic force exerted on moving electric charges, electric currents, or magnetic materials.
In the case of the conductive wire, the magnetic field is produced by the moving charges and is proportional to the current.
Fig. 2 - A conductive wire with a magnetic field.
We can distinguish three separate cases of magnetic fields when it comes to conductive wires, as pictured in Figure 2 above.
Inside the wire, the magnetic field is equal to
\[B=\frac{\mu_0 J r}{2},\]
where we can reexpress the current density as
\[J=\frac{I}{A}= \frac{I}{\pi R^2},\]
so the final expression becomes
\[B=\frac{\mu_0 r I}{2\pi R^2}.\]
As we get further away from the center of the wire and reach its surface, the equation transforms into
\[B=\frac{\mu_0 I}{2\pi R}.\]
Finally, on the outside, the magnetic field is equal to
\[B=\frac{\mu_0 I}{2\pi r'}.\]
Here, \(\mu_0\) is the vacuum permeability equal to \(4\pi\times10^{-7} \, \frac{\mathrm{T} \, \mathrm{m}}{\mathrm{A}}\).
A common concept in electrochemistry is the exchange current density.
Exchange current density occurs when the current density is flowing in two opposite directions simultaneously.
These two opposing currents are called anodic and cathodic. As a result, the net current density is zero, as the anodic and cathodic currents cancel each other out. These values must be obtained experimentally, as it's the main way to quantify the performance of electrodes.
Fig. 3 - Exchange current density can be used to quantify the properties of electrodes.
The electric field in a wire is proportional to the resistivity of the conductor and the current density: (\vec{J}=\frac{\vec{E}}{\rho}.\)
Magnetic field is a vector field that describes the magnetic force exerted on moving electric charges, electric currents, or magnetic materials.
In the case of the conductive wire, the magnetic field is produced by the moving charges and is proportional to the current.
Exchange current density occurs when the current density is flowing in two opposite directions simultaneously.
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How to calculate current density?
Current density can be calculated using J=I/A.
What is current density?
Current density is the flow of charge per unit area.
How to measure current density of an electrode?
Current density of an electrode can be measured using anodic and cathodic currents.
What is the unit of current density?
The unit of current density is amperes per meter squared.
What does current density depend on?
Current density depends on the current and the cross-sectional area of the wire.
Why do we need current density?
We need current density to explain and quantify the microscopic properties of current.
What does high current density mean?
High current density means reduced cell voltage.
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