# Steady Currents and Circuits

Physics II · Electromagnetism · https://tryals.app/learn/physics-ii/steady-currents-and-circuits

## Charge That Keeps Moving

A **current density** $\mathbf{J}$ carries charge, and charge conservation is the **continuity equation**:

$$\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t}$$

For **steady** currents nothing accumulates anywhere, so $\nabla \cdot \mathbf{J} = 0$, current in equals current out at every junction. That is Kirchhoff's current law, stated as a field equation.

In an **ohmic** conductor the response is linear: $\mathbf{J} = \sigma \mathbf{E}$, with $\sigma$ the conductivity and $\rho_{res} = 1/\sigma$ the resistivity. Integrating across a uniform wire gives the familiar

$$R = \frac{\rho_{res} L}{A}, \qquad V = IR$$

Resistance is geometry plus material, long and thin resists, short and fat conducts.

A steady current in a closed loop cannot be driven by an electrostatic field, because $\oint \mathbf{E}\cdot d\mathbf{l} = 0$ means no net energy per loop. Something else must do the work: a **generator** supplying a non-electrostatic field, whose line integral is the **electromotive force**

$$\mathcal{E} = \oint \mathbf{E}_{motor} \cdot d\mathbf{l}$$

EMF is measured in volts but is not a potential difference, it is work per unit charge delivered by a chemical, mechanical or magnetic agent. A real source has internal resistance $r$, so its terminal voltage is $\mathcal{E} - Ir$, always below the EMF when delivering current.

The energy balance in a circuit is exact: the source supplies $\mathcal{E}I$, of which $I^2r$ heats the source itself and the rest reaches the load. **Joule heating** $P = I^2R = V^2/R$ is irreversible, the ordered drift energy ends up as random thermal motion.

> **Common pitfall:** treating EMF and terminal voltage as the same number. They coincide only at zero current. Under load the terminal voltage sags by $Ir$, which is why a failing battery reads fine unloaded and collapses the moment it has to deliver.

## Practice questions

6 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. A copper wire has resistivity $1.7 \times 10^{-8}$ Ω m, length 2.0 m and cross-sectional area $1.0 \times 10^{-6}$ m$^2$. Compute its resistance in mΩ, to the nearest whole number.

**Answer:** 34 (within ±1)

**Why:** $R = \rho L/A = 1.7 \times 10^{-8} \times 2.0/10^{-6} = 3.4 \times 10^{-2}\ \Omega = 34$ mΩ. Copper’s tiny resistivity is why household wiring can be thin and still lossless enough.

Page: https://tryals.app/practice/physics-ii/steady-currents-and-circuits/a-copper-wire-has-resistivity-1-7-10-m-length-2-0-m-and

### 2. A steady current requires a non-zero current density everywhere in a closed loop, yet electrostatic fields satisfy the conservative condition $\oint \mathbf{E}\cdot d\mathbf{l} = 0$. What physical consequence follows for sustained circulation?

A. A non-electrostatic field must perform net work along the loop
B. The electric field must vanish identically inside the conductor
C. Charge accumulation at junctions drives the circulating carriers
D. Resistance must drop to zero for steady current to be maintained

**Answer:** A. A non-electrostatic field must perform net work along the loop

**Why:** Electrostatic fields cannot sustain circulation because their closed line integral vanishes, meaning charges gain no net energy over a complete circuit without a non-conservative source. Confusing steady currents with zero resistance or zero internal field ignores how Ohm's law and dissipation operate.

Page: https://tryals.app/practice/physics-ii/steady-currents-and-circuits/a-steady-current-requires-a-non-zero-current-density-everywhere-in-a

### 3. The terminal voltage of a real battery equals its EMF only when it delivers no current.

**Answer:** True

**Why:** True, the terminal voltage is $\mathcal{E} - Ir$, which equals $\mathcal{E}$ only at $I = 0$. This is why a tired battery reads fine on an unloaded meter and collapses under load.

Page: https://tryals.app/practice/physics-ii/steady-currents-and-circuits/the-terminal-voltage-of-a-real-battery-equals-its-emf-only-when-it

### 4. A resistor of 8 Ω carries a current of 2.5 A. Compute the power dissipated in watts.

**Answer:** 50 (within ±0.5)

**Why:** $P = I^2R = 6.25 \times 8 = 50$ W. The quadratic dependence on current is why transmission lines run at high voltage and low current.

Page: https://tryals.app/practice/physics-ii/steady-currents-and-circuits/a-resistor-of-8-carries-a-current-of-2-5-a-compute-the-power

### 5. Which statements about steady currents are correct?

A. Steady currents require charge to accumulate somewhere
B. The divergence of the current density is zero
C. Current into a junction equals current out of it
D. Ohmic conductors satisfy J equals sigma times E

**Answer:** B. The divergence of the current density is zero; C. Current into a junction equals current out of it; D. Ohmic conductors satisfy J equals sigma times E

**Why:** Steady means nothing accumulates, so $\nabla\cdot\mathbf{J} = 0$, which is Kirchhoff’s junction rule in field form. Accumulation is exactly what steadiness forbids.

Page: https://tryals.app/practice/physics-ii/steady-currents-and-circuits/which-statements-about-steady-currents-are-correct

### 6. Match each quantity to its definition.

**Answer:**

- Current density → Charge flow per unit area per unit time
- Conductivity → The constant relating current density to field
- Electromotive force → Work per unit charge from a non-electrostatic source
- Internal resistance → What makes terminal voltage sag under load

**Why:** Current density and conductivity describe the medium; EMF is the work a generator supplies per unit charge; and internal resistance is why a loaded source delivers less than its EMF.

Page: https://tryals.app/practice/physics-ii/steady-currents-and-circuits/match-each-quantity-to-its-definition