Question

The dimensions of a conductor of specific resistance $\rho$ are shown below: What is its resistance across $\mathrm{CD}$. (A) $(\mathrm{pb}/\mathrm{ac})$ (B) ( $\rho \mathrm{a}/\mathrm{bc})$ (C) $[(\rho \mathrm{ab})/\mathrm{c}]$ (D) $(\rho c/ab)$

Short answer

The correct option for the resistance across CD is (D) $(\rho c/ab)$.

Step-by-step solution

Recall the formula for resistance

Recall that the formula for resistance (R) in terms of resistivity (ρ), length (L) and area of cross-section (A) is given by: $$R=\frac{\rho \cdot L}{A}$$ Now, we will apply this formula to the given problem.

Identify dimensions of the conductor

The given conductor has the following dimensions: Length (L) = CD Width (W) = AB Thickness (T) = AC The area of cross-section is the product of the width and the thickness, which is equal to: $$A=W\cdot T=\mathrm{AB}\cdot \mathrm{AC}$$

Calculate the resistance

We can now calculate the resistance across CD by substituting values from Step 2 into the formula mentioned in Step 1: $$R=\frac{\rho \cdot (\mathrm{CD})}{\mathrm{AB}\cdot \mathrm{AC}}$$

Compare options

Now, let's compare this expression of resistance with the multiple-choice options provided: (A) $(\mathrm{pb}/\mathrm{ac})$: This option does not match the expression obtained in Step 3. (B) $(\rho \mathrm{a}/\mathrm{bc})$: This option does not match the expression obtained in Step 3. (C) $[(\rho \mathrm{ab})/\mathrm{c}]$: This option also does not match the expression obtained in Step 3. (D) $(\rho c/ab)$: By assigning: $\mathrm{CD}=c$ We can rewrite our expression from Step 3 as: $$R=\frac{\rho \cdot c}{a\cdot b}$$ This option matches the expression obtained in Step 3.

Choose correct option

As we have compared the expression of resistance received in Step 3 with the given options, we can conclude that the correct option is: (D) $(\rho c/ab)$

Key concepts

Resistivity

Resistivity, represented by the Greek letter $\rho$, plays a vital role in determining how much a material opposes the flow of electric current. Resistivity is an intrinsic property of a material, which means it is a characteristic that depends only on the type of material and its temperature, rather than its shape or size. The formula that involves resistivity to find the resistance $R$ of a conductor is given by: $$R=\frac{\rho \cdot L}{A}$$ where $L$ is the length of the conductor and $A$ is the cross-sectional area. A higher resistivity indicates a stronger opposition to current flow, making materials like rubber excellent insulators, while metals like copper, with low resistivity, are excellent conductors.

• Resistivity depends on material type and temperature.
• It is an intrinsic property, independent of the dimensions.
• Materials with low resistivity are good conductors.

Cross-sectional area

The cross-sectional area $A$ of a conductor is the area of its face that is perpendicular to the current flow. It plays a critical role in determining the resistance of the conductor. The larger the cross-sectional area, the less resistance the conductor provides, allowing more current to flow. This reduction in resistance with increased area can be understood since a larger area provides more paths for the current particles to move through, easing the current flow. In our specific exercise, the cross-sectional area $A$ is given by the product of the width $AB$ and thickness $AC$: $$A=AB\cdot AC$$

• Increasing cross-sectional area decreases the resistance.
• The area is perpendicular to the direction of current flow.
• It is calculated by the product of two perpendicular dimensions.

Length of conductor

The length $L$ of a conductor is the distance between the points where the current enters and leaves the conductor. In the context of resistance, the length directly affects the amount of material the current must pass through, influencing the resistance value. When using the equation $R=\frac{\rho \cdot L}{A}$, an increase in length $L$ increases the resistance. This is because a longer conductor means that the electric current has more material to travel through, which usually results in more collisions and, hence, higher resistance. In the exercise, the length corresponds to the dimension $CD$ of the conductor.

• Length is directly proportional to resistance.
• Shorter conductors have less resistance.
• Electric current faces more resistance in longer conductors.