Question

(a) What is the unknown${\mathbf{emf}}_{\mathbf{x}}$in a potentiometer that balances when${\mathbf{R}}_{\mathbf{x}}$is $\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\Omega }$ , and balances when role="math" localid="1656393398590" ${\mathbf{R}}_{\mathbf{s}}$is $\mathbf{15}\mathbf{.}\mathbf{0}\mathbf{\Omega }$for a standard 3.000-V emf? (b) The same${\mathbf{emf}}_{\mathbf{x}}$is placed in the same potentiometer, which now balances when ${\mathbf{R}}_{\mathbf{s}}$is $\mathbf{15}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\Omega }$ for a standard emf of 3.100-V . At what resistance${\mathbf{R}}_{\mathbf{x}}$will the potentiometer balance?

Short answer

The unknown emf of the potentiometer is ${\mathrm{E}}_{\mathrm{x}}=2\mathrm{V}$.

(b) The unknown resistance for the potentiometer is ${\mathrm{R}}_{\mathrm{x}}=9.7\mathrm{\Omega }$.

Step-by-step solution

Concept Introduction

The rate of electron flow can be described as the aggregate flow of electrons via a wire. The term "resistance" refers to anything that stands in the way of current flow. To convert electrical energy to light, heat, or movement, an electrical circuit must have resistance.

The electric potential generated by an electrochemical cell or a changing magnetic field is known as electromotive force. Electromotive force (EMF) is a well-known abbreviation.

Emf in a potentiometer(a)

Calculate the emf ${\mathrm{E}}_{\mathrm{x}}$in a potentiometer that balances when $R_x=10\Omega$and ${\mathrm{R}}_{\mathrm{s}}=15\mathrm{\Omega }$ for standard emf ${\mathrm{E}}_{\mathrm{s}}=3\mathrm{V}$. Now, it can be calculated -

role="math" localid="1656395116969" $$\begin{gathered} {\mathrm{E}}_{\mathrm{x}}={\mathrm{E}}_{\mathrm{s}}\frac{{\mathrm{R}}_{\mathrm{x}}}{{\mathrm{R}}_{\mathrm{s}}} \\ =(3\mathrm{V})\frac{10\mathrm{\Omega }}{15\mathrm{\Omega }} \\ =2\mathrm{V} \end{gathered}$$

Therefore, the emf value is obtained as ${\mathrm{E}}_{\mathrm{x}}=2\mathrm{V}$.

Resistance in a potentiometer(b)

Connect the same emf ${\mathrm{E}}_{\mathrm{x}}$, in a potentiometer that balances when ${\mathrm{R}}_{\mathrm{s}}=15\mathrm{\Omega }$for standard emf . The resistance is then,

$$\begin{gathered} {\mathrm{E}}_{\mathrm{x}}={\mathrm{E}}_{\mathrm{s}}\frac{{\mathrm{E}}_{\mathrm{x}}}{{\mathrm{E}}_{\mathrm{s}}} \\ =(15\Omega )\frac{2\mathrm{V}}{3.1V} \\ =9.7\Omega \end{gathered}$$

Therefore, the value for resistance is ${\mathrm{R}}_{\mathrm{x}}=9.7\mathrm{\Omega }$.