Question

If \(\sigma_{1}, \sigma_{2}\), and \(\sigma_{3}\) are the conductance's of three conductor then equivalent conductance when they are joined in series, will be. (A) \(\sigma_{1}+\sigma_{2}+\sigma_{3}\) (B) \(\left(1 / \sigma_{1}\right)+\left(1 / \sigma_{2}\right)+\left(1 / \sigma_{3}\right)\) (C) \(\left\\{\left(\sigma_{1} \sigma_{2} \sigma_{3}\right) /\left(\sigma_{1}+\sigma_{2}+\sigma_{3}\right)\right\\}\) (D) None of these.

Short answer

The equivalent conductance when they are joined in series is given by: $$ \sigma_{eq} = \frac{1}{\left( \frac{1}{\sigma_{1}} + \frac{1}{\sigma_{2}} + \frac{1}{\sigma_{3}} \right)} $$ Hence, the correct answer is (B).

Step-by-step solution

Understand the reciprocal property between resistance and conductance

Conductance (\(\sigma\)) is the reciprocal of resistance (R). Mathematically, this can be expressed as: $$ \sigma = \frac{1}{R} $$ And the resistance can be expressed as: $$ R = \frac{1}{\sigma} $$

Find the equivalent resistance for resistors connected in series

When resistors are connected in series, their equivalent resistance (R_eq) can be found by adding the individual resistances: $$ R_{eq} = R_{1} + R_{2} + R_{3} $$

Convert conductances to resistances

Using the reciprocal property, we can find the resistances corresponding to the given conductances: $$ R_{1} = \frac{1}{\sigma_{1}}, \quad R_{2} = \frac{1}{\sigma_{2}}, \quad R_{3} = \frac{1}{\sigma_{3}} $$

Plug in the expressions for the resistances into the series formula

Plug in the resistances found in step 3 into the formula for equivalent resistance for resistors connected in series: $$ R_{eq} = \frac{1}{\sigma_{1}} + \frac{1}{\sigma_{2}} + \frac{1}{\sigma_{3}} $$

Convert the equivalent resistance to equivalent conductance

Now, we will convert the equivalent resistance back to the equivalent conductance using the reciprocal property: $$ \sigma_{eq} = \frac{1}{R_{eq}} $$ Plugging in the expression for \(R_{eq}\) found in step 4: $$ \sigma_{eq} = \frac{1}{\left( \frac{1}{\sigma_{1}} + \frac{1}{\sigma_{2}} + \frac{1}{\sigma_{3}} \right)} $$

Choose the correct answer

The resulting expression for the equivalent conductance is: $$ \sigma_{eq} = \frac{1}{\left( \frac{1}{\sigma_{1}} + \frac{1}{\sigma_{2}} + \frac{1}{\sigma_{3}} \right)} $$ Comparing our result from step 5 with the given options: (A) \(\sigma_{1}+\sigma_{2}+\sigma_{3}\) (B) \(\left(1 / \sigma_{1}\right)+\left(1 / \sigma_{2}\right)+\left(1 /\sigma_{3}\right)\) (C) \(\left\\{\left(\sigma_{1} \sigma_{2} \sigma_{3}\right)/\left(\sigma_{1}+\sigma_{2}+\sigma_{3}\right)\right\\}\) (D) None of these. We can observe that our result corresponds to option (B), which is the correct answer.

Key concepts

Conductors in Series

In the world of electricity, conductors are essential components that allow electric current to flow from one point to another. In many electrical systems, multiple conductors are combined in series to enable the flow of electricity. When conductors are arranged in series, it means they are connected one after another in a single pathway.

Understanding how conductance works in a series configuration is important. Conductance (\(\sigma\)) measures how easily electricity flows through a material. It is the reciprocal of resistance, which indicates how much a material opposes the flow of current.

When conductors are connected in series, their equivalent conductance is not simply the sum of their individual conductances. Instead, series conductance follows a specific rule. To find the equivalent conductance, you need to take the reciprocal of the sum of reciprocals of individual conductances. This is different from connecting in parallel, where you would just sum them up directly.

Reciprocal Relations in Electricity

In electrical systems, the concept of reciprocal relations allows for a deeper understanding of the relationship between resistance and conductance. Conductance (\( \sigma \)) and resistance (\( R \)) are intrinsically linked. They represent two sides of the same coin.

Mathematically, they are related through reciprocal relations. Conductance is defined as the reciprocal of resistance:\[\sigma = \frac{1}{R}\]
Similarly, resistance is the reciprocal of conductance:\[R = \frac{1}{\sigma}\]
This reciprocal relationship is crucial when dealing with conductors in series, as it affects how we compute the equivalent resistance and therefore, the equivalent conductance. It’s through understanding these reciprocal relations that one can convert resistance values back and forth between conductance, aiding in tasks such as simplifying series and parallel networks.

Equivalent Conductance

Equivalent conductance is a key concept in electrical circuits, especially when dealing with multiple conductors. It represents a single conductance value that can replace a series of conductors with the same effect on the flow of electric current.

For conductors connected in series, to find the equivalent conductance, the process involves several steps due to the reciprocal nature of conductance and resistance. First, you must determine the resistance of each conductor using the formula:\[ R_i = \frac{1}{\sigma_i} \]
Next, calculate the total equivalent resistance by summing the individual resistances:\[R_{eq} = R_1 + R_2 + R_3 \]
Finally, convert this equivalent resistance back to equivalent conductance by taking the reciprocal:\[\sigma_{eq} = \frac{1}{R_{eq}} \]
This gives you the equivalent conductance of the series combination of conductors. Understanding this process is essential in quickly analyzing and designing efficient electrical systems.