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Ohm's Law

CSCA Ohm's Law study guide organized around the publicly available CSCA syllabus. Practice Physics questions on aicsca.com.

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Topic: Electromagnetism > DC Circuits > Ohm's Law

1. Core Law: Ohm's Law

Ohm's Law is the cornerstone of circuit analysis. It reveals the quantitative relationship between voltage, current, and resistance in **pure resistive elements**.

* **Formula**: $$I = \frac{U}{R}$$

* **Physical Meaning**: The current $I$ flowing through a conductor is directly proportional to the voltage $U$ across it and inversely proportional to its resistance $R$.

* **Scope of Application**: **Pure Resistive Circuits** (e.g., metal wires, resistors, electrolytes).

* *Note: It does not fully apply to non-linear components (like diodes, gaseous conductors) or active circuits containing EMF (like running motors).*

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2. The "Two Faces" of Resistance

In the CSCA exam, distinguishing between the **Definition Formula** and the **Determination Formula** is a frequent testing point.

| Type | Formula | Significance | Notes |

| :--- | :---: | :--- | :--- |

| **Definition** | $$R = \frac{U}{I}$$ | Provides a method to **measure** resistance. | **$R$ is independent of $U$ and $I$**. For a fixed resistor, $R$ exists even if $U=0$. |

| **Determination** | $$R = \rho \frac{L}{S}$$ | Reveals factors that **determine** resistance. | $\rho$: Resistivity (Material/Temp)<br>$L$: Length<br>$S$: Cross-sectional Area |

**Common Mistake Warning**:

Many students think "$R$ is proportional to $U$ and inversely proportional to $I$". This is **WRONG**! Resistance is an intrinsic property determined by material and geometry (Determination Formula), not by voltage or current (Definition Formula).

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3. Volt-Ampere Characteristic Curve ($I-U$ Graph)

Analyzing component properties via graphs is crucial for physics competitions and exams.

* **Linear Element (Ohmic)**: The graph is a straight line passing through the origin. Slope $k = \frac{I}{U} = \frac{1}{R}$. Larger slope means smaller resistance.

* **Non-linear Element (Non-Ohmic)**: The graph is a curve (e.g., light bulb filament, diode). Resistance changes with voltage/current. In this case, $R = U/I$ calculates the resistance at that specific operating point.

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4. Key to Problem Solving: Consistency Principle

When applying $I = U/R$, you must ensure that **$I$, $U$, and $R$ belong to the SAME object** (the same conductor or the same component).

Typical Example Analysis

**Example: Mixed Circuit Calculation**

**Problem**: In the circuit shown, the source voltage is $U = 12V$. Resistors are $R_1 = 3\Omega$, $R_2 = 6\Omega$, and $R_3 = 6\Omega$. $R_2$ and $R_3$ are connected in parallel, and this combination is connected in series with $R_1$. Find: (1) Total resistance; (2) Current through $R_1$; (3) Voltage across $R_2$.

**Solution Steps**:

1. **Analyze Structure**: $R_2 // R_3$ (Parallel), then $+ R_1$ (Series).

2. **Total Resistance**:

* Parallel part: $\frac{1}{R_{23}} = \frac{1}{R_2} + \frac{1}{R_3} = \frac{1}{6} + \frac{1}{6} \Rightarrow R_{23} = 3\Omega$

* Total: $R_{tot} = R_1 + R_{23} = 3 + 3 = 6\Omega$

3. **Total Current ($I_1$)**:

* Ohm's Law (Whole circuit): $I_{tot} = \frac{U}{R_{tot}} = \frac{12V}{6\Omega} = 2A$

* Current is the same in series, so current through $R_1$ is $I_1 = 2A$.

4. **Voltage across $R_2$ ($U_2$)**:

* **Method 1 (Ohm's Law on Part)**: Calculate voltage across the parallel block. $U_{23} = I_{tot} \times R_{23} = 2A \times 3\Omega = 6V$. Since parallel voltages are equal, $U_2 = 6V$.

* **Method 2 (Potential Difference)**: Voltage drop on $R_1$ is $U_1 = I_1 R_1 = 6V$. Remaining voltage for the parallel part is $U_2 = U - U_1 = 6V$.

**Answer**: (1) $6\Omega$; (2) $2A$; (3) $6V$.