Understanding the Relationship Between Current, Resistance, and Voltage

The question asks why the current in a conductor halves when its resistance is doubled. This relationship is governed by a fundamental law in electricity known as Ohm's Law.

What is Ohm's Law?

Ohm's Law describes the relationship between voltage (V), current (I), and resistance (R) in a conductor. It states that the current through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance between them, provided the temperature and other physical conditions remain constant.

Mathematically, Ohm's Law is expressed as:

\begin{equation*} \text{V} = \text{I} \times \text{R} \end{equation*}

This formula can be rearranged to solve for current (I), voltage (V), or resistance (R):

• Current: \begin{equation*} \text{I} = \frac{\text{V}}{\text{R}} \end{equation*}
• Voltage: \begin{equation*} \text{V} = \text{I} \times \text{R} \end{equation*}
• Resistance: \begin{equation*} \text{R} = \frac{\text{V}}{\text{I}} \end{equation*}

Analyzing the Relationship

The question specifically focuses on the relationship between current (I) and resistance (R) when the voltage (V) is constant. From the formula $\text{I} = \frac{\text{V}}{\text{R}}$, we can see that:

• If V is kept constant, I is inversely proportional to R. This means that if R increases, I decreases, and if R decreases, I increases.
• The extent of the change is proportional. If R is doubled, and V is constant, then I must be halved to maintain the equality \begin{equation*} \text{V} = \text{I} \times \text{R} \end{equation*}.

Let's consider an example:

Suppose a conductor has a resistance \(\text{R}_1 = 10 \Omega\) and a voltage \(\text{V} = 20 \text{ V}\) is applied across it.

The current flowing through it will be \begin{equation*} \text{I}_1 = \frac{\text{V}}{\text{R}_1} = \frac{20 \text{ V}}{10 \Omega} = 2 \text{ A} \end{equation*}

Now, if the resistance is doubled to \(\text{R}_2 = 2 \times \text{R}_1 = 2 \times 10 \Omega = 20 \Omega\), and the voltage remains the same (\(\text{V} = 20 \text{ V}\)), the new current \(\text{I}_2\) will be:

\begin{equation*} \text{I}_2 = \frac{\text{V}}{\text{R}_2} = \frac{20 \text{ V}}{20 \Omega} = 1 \text{ A} \end{equation*}

As you can see, when the resistance doubled (from 10 \(\Omega\) to 20 \(\Omega\)), the current halved (from 2 A to 1 A), assuming the voltage remained constant.

Evaluating the Options

The question provides several possible formulas:

1. \(\text{I} = \text{V} - \text{R}\): This formula is incorrect. Subtracting resistance from voltage does not give current.
2. \(\text{I} = \text{V}/\text{R}\): This is the correct expression of Ohm's Law when solving for current. It shows the inverse relationship between current and resistance, which explains why doubling resistance halves the current (for a constant voltage).
3. \(\text{I} = (\text{R}/\text{V})\text{n}\): This formula is incorrect and does not represent the relationship between current, voltage, and resistance in Ohm's Law. The 'n' is not defined in this context.
4. \(\text{I} = \text{V R}\): This formula suggests that current is directly proportional to the product of voltage and resistance, which contradicts Ohm's Law and the observed inverse relationship between current and resistance.

Therefore, the reason why the current gets halved when the resistance of a conductor is doubled (assuming voltage is constant) is because of the relationship described by Ohm's Law, which is $\text{I} = \text{V}/\text{R}$.

Revision Table: Ohm's Law Concepts

Additional Information: Factors Affecting Resistance and Ohm's Law Applications

While Ohm's Law provides a fundamental relationship, it's important to know that the resistance of a conductor itself can be influenced by several factors:

• Material: Different materials have different inherent resistances (resistivity). Conductors like copper and aluminum have low resistance, while insulators like rubber and glass have very high resistance.
• Length: The resistance of a conductor is directly proportional to its length. A longer wire has more resistance.
• Cross-sectional Area: The resistance of a conductor is inversely proportional to its cross-sectional area. A thicker wire has less resistance.
• Temperature: For most conductors, resistance increases with increasing temperature.

Ohm's Law is crucial for understanding and designing electrical circuits. It is used in:

• Calculating the current flowing through a component.
• Determining the voltage drop across a resistor.
• Calculating the required resistance for a specific current or voltage.
• Troubleshooting electrical problems.

Understanding the simple formula $\text{I} = \text{V}/\text{R}$ is key to grasping many concepts in introductory electricity.