11.1 Electric Current

Electric current is the rate at which charge flows through a cross-sectional area, written as $I = \frac{dq}{dt}$. Inside a conductor, that current comes from charge carriers drifting with an average velocity, so you can also write $I = nqv_d A$ and connect it to current density $\vec{J} = nq\vec{v}_d$.

Why This Matters for the AP Physics C: E&M Exam

Unit 11 carries one of the largest weightings on the AP Physics C: E&M exam, and electric current is the foundation the rest of the unit builds on. Once you understand how charge moves through a conductor, you can reason about resistance, power, Kirchhoff's rules, and RC circuits more confidently.

This topic supports several kinds of exam thinking:

• Deriving symbolic expressions, such as connecting $I$, $n$, $q$, $v_d$, and $A$, or building $I = \int \vec{J} \cdot d\vec{A}$ from a given current density function.
• Predicting how current changes when one variable changes, using the functional dependence in $I = nqv_d A$.
• Applying definitions and relationships to make and support claims about charge movement.
• Creating and reading diagrams that represent how charge flows through a conductor.

The lab-style free response question in this course asks you to design experiments, linearize data, and analyze results. A clean understanding of current and current density gives you the vocabulary and models you need when current shows up in those experimental setups.

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Key Takeaways

• Current is the rate of charge flow through a cross-sectional area: $I = \frac{dq}{dt}$, measured in amperes (coulombs per second).
• Inside a conductor, current depends on carrier density $n$, charge per carrier $q$, drift velocity $v_d$, and area $A$: $I = nqv_d A$.
• Current density $\vec{J} = nq\vec{v}_d$ is a vector; total current is the surface integral $I = \int \vec{J} \cdot d\vec{A}$.
• A potential difference creates an electric field inside the conductor, related to current density by $\vec{E} = \rho\vec{J}$.
• Conventional current points the way positive charge would move, even though electrons (negative carriers) usually do the actual moving.
• Current has a direction but is a scalar, so it does not add like a vector or split into x, y, z components.

Core Concepts

Current Through a Cross-Sectional Area

Current is the rate at which electric charge passes through a cross-sectional area of a conductor:

$I=\frac{dq}{dt}$

This definition works for any situation where charge flows. Charges move in response to an electric potential difference, sometimes called electromotive force or emf ($\mathcal{E}$), which provides the push that drives charge through a circuit.

Looking inside the wire, current happens because charge carriers travel through the conductor with an average drift velocity. You can write the current through a cross-sectional area $A$ as:

$I=n q v_{d} A$

Where:

• $n$ is the charge carrier density (carriers per unit volume)
• $q$ is the charge of each carrier
• $v_d$ is the drift velocity of the carriers
• $A$ is the cross-sectional area of the conductor

If the current is zero in a section of wire, the net motion of charge carriers there is also zero. Individual carriers are still moving, just randomly, so their average motion cancels out. Think of water molecules in a still pond moving randomly with no overall flow.

Current Density

Current density describes the flow of charge per unit area inside a conductor, which tells you how current spreads across the cross-section.

The current density vector is:

$\vec{J}=n q \vec{v}_{d}$

It connects to total current through a surface by:

$I=\int \vec{J} \cdot d\vec{A}$

where $d\vec{A}$ is an area element pointing perpendicular to the surface.

A potential difference across a conductor creates an electric field inside it. That field is proportional to both the resistivity $\rho$ and the current density $\vec{J}$:

$\vec{E}=\rho \vec{J}$

Current density is a vector, so it has both magnitude and direction. It points along the charge flow, and its magnitude tells you how much charge passes per unit area per unit time. In a uniform conductor the current density is usually constant across the cross-section, but in a non-uniform conductor it can vary with position.

Total Current from Current Density

When current density varies across the cross-section, find the total current by integrating over the whole area:

$I_{\text{tot}}=\int \vec{J}(r) \cdot d \vec{A}$

This handles flow that is not uniform, which is useful for conductors with varying geometry or material properties.

Direction of Current

Current has a defined direction, but it is not a vector in the usual sense. By convention, current direction is the direction positive charge would move.

In most circuits, electrons carry the charge and they are negative, so:

• Conventional current flows from the positive terminal to the negative terminal.
• Electron flow goes the opposite way, from negative to positive.

Even though current has a direction, it is a scalar. Its direction is defined relative to the charge carriers, not relative to spatial axes, so current does not follow the rules of vector addition and has no x, y, or z components.

How to Use This on the AP Physics C: E&M Exam

Problem Solving

• When you know carrier properties, use $I = nqv_d A$ directly. Keep units consistent: $n$ in carriers per cubic meter, $q$ in coulombs, $v_d$ in meters per second, $A$ in square meters.
• For uniform current spread over an area, current density is simply $J = I/A$. Compute the area carefully, especially for a circular wire where $A = \pi r^2$.
• When a problem gives $\vec{J}$ as a function of position, set up $I = \int \vec{J} \cdot d\vec{A}$ and choose an area element that matches the symmetry (often a thin ring for cylindrical wires).

Free Response

• Show your reasoning, not just the final number. Define each variable and state which relationship you are applying before plugging in.
• If asked to predict a factor of change, use the functional dependence. For example, doubling drift velocity in $I = nqv_d A$ doubles the current if everything else stays fixed.
• For lab-style questions, be ready to describe how you would measure current or relate it to drift speed, and how you would linearize and graph data to support a claim.

Common Trap

• Watch the sign and direction setup: conventional current and electron flow point opposite ways. Use conventional current unless a problem clearly tells you otherwise.

Practice Problem 1: Current Calculation

Solution

Use: $I = nqv_dA$

Where:

• $n = 8.5 \times 10^{28} \text{ electrons/m}^3$
• $q = 1.6 \times 10^{-19} \text{ C}$ (magnitude of electron charge)
• $v_d = 3.0 \times 10^{-4} \text{ m/s}$
• $A = 2.0 \times 10^{-6} \text{ m}^2$

Substituting: $I = (8.5 \times 10^{28})(1.6 \times 10^{-19})(3.0 \times 10^{-4})(2.0 \times 10^{-6})$

$I = 8.16 \text{ A}$

The current in the wire is about 8.16 amperes.

Practice Problem 2: Current Density

Solution

Current density is current divided by area: $J = \frac{I}{A}$

First find the cross-sectional area: $A = \pi r^2 = \pi (1.2 \times 10^{-3} \text{ m})^2 = 4.52 \times 10^{-6} \text{ m}^2$

Then: $J = \frac{5.0 \text{ A}}{4.52 \times 10^{-6} \text{ m}^2} = 1.11 \times 10^6 \text{ A/m}^2$

The current density is $1.11 \times 10^6 \text{ A/m}^2$.

Common Misconceptions

• "Current is a vector." Current has a direction but is a scalar. It does not add like a vector or break into components. Current density $\vec{J}$, on the other hand, is a true vector.
• "Charge carriers move fast through the wire." Drift velocity is usually very slow (often well under a millimeter per second). The electric field, not the carriers themselves, propagates quickly through the circuit.
• "Zero current means carriers are frozen." Zero current just means no net motion. Individual carriers still move randomly at nonzero speeds; their motions cancel.
• "Electrons flow in the direction of conventional current." In typical circuits the carriers are electrons, which move opposite to conventional current. Conventional current is defined by where positive charge would go.
• "Current density is always uniform." It can vary across the cross-section. When it does, you must integrate $\vec{J} \cdot d\vec{A}$ rather than just divide current by area.
• "$\vec{E} = \rho\vec{J}$ only applies in empty space." This relationship describes the field inside a conducting material, linking the field, the material's resistivity, and the current density.

Related AP Physics C: E&M Guides

• 11.2 Electric Circuits
• 11.3 Resistance, Resistivity, and Ohm's Law
• 11.5 Compound Direct Current Circuits
• 11.8 Resistor-Capacitor (RC) Circuits
• 11.4 Electric Power
• 11.6 Kirchhoff's Loop Rule