Answer
Verified
399.5k+ views
Hint: When the galvanometer shows no deflection, there is zero current in the galvanometer. This happens when the potential difference and emf of an unknown cell are equal. Using V=iR, find the current in the loop ABD and then the value of resistance across AJ in terms of r. Later, use $R=\dfrac{\rho l}{A}$ and find the relation between length AJ and L.
Formula used:
V=iR
$R=\dfrac{\rho l}{A}$
Complete step by step solution:
A potentiometer is a device that is used to measure the emf of an unknown cell. You can see a simple setup of a potentiometer in the figure given in the question.
We know that a cell creates a potential difference across a wire or a resistor. And due to resistance of the wire, there is a flow of charges and we say that there is a current in the circuit.
Therefore, there will be some current in the given circuit. A galvanometer is a device used to detect current in the circuit. Hence, if a current is flowing through the galvanometer, it will show some readings.
When the galvanometer shows no deflection or zero reading, there is no current flowing through the galvanometer. We say that the circuit is balanced and the length AJ is called balancing length. This is possible when the potential difference across the AJ is equal to the emf of the cell C.
Let the potential difference across AJ be V.
Hence, $V=\dfrac{E}{2}$
There will be a current in the loop consisting points A, B and D. Let that current be i. The total resistance in the loop is 12r+r = 13r. From Ohm’s law, we know V=iR. Hence, E=i(13r).
$\Rightarrow i=\dfrac{E}{13r}$.
Let the balancing length (AJ) be l and resistance of AJ be R’.
The potential difference across AJ is $V=\dfrac{E}{2}$. The current in AJ is $i=\dfrac{E}{13r}$.
Hence, V=iR’
$\Rightarrow \dfrac{E}{2}=\left( \dfrac{E}{13r} \right)R'$
$\Rightarrow \dfrac{1}{2}=\dfrac{R'}{13r}$
$\Rightarrow R'=\dfrac{13r}{2}$
The cross sectional area of wire AB be uniform and be A. Let its resistivity be $\rho $.
Therefore, $R'=\dfrac{\rho l}{A}$ ….. (i)
And
$12r=\dfrac{\rho L}{A}$ ….. (ii)
Divide equation (i) and equation (ii).
$\Rightarrow \dfrac{R'}{12r}=\dfrac{\dfrac{\rho l}{A}}{\dfrac{\rho L}{A}}$
$\Rightarrow \dfrac{R'}{12r}=\dfrac{l}{L}$
$\Rightarrow R'=\dfrac{12rl}{L}$
And we found that $R'=\dfrac{13r}{2}$
This means that $\dfrac{13r}{2}=\dfrac{12rl}{L}$
$\Rightarrow l=\dfrac{13L}{24}$
The balancing length is $\dfrac{13L}{24}$,
Hence, the correct option is D.
Note: Note that the emf of the unknown cell (i.e C) must be less than the emf of the main cell (D). Otherwise, we will not find a balancing length.
The emf of the unknown cell is given by $E=\dfrac{Vl}{L}$, where V is the potential difference across the resistance wire (AB).
The drawback of this method is that we cannot find the internal resistance of the unknown cell.
Formula used:
V=iR
$R=\dfrac{\rho l}{A}$
Complete step by step solution:
A potentiometer is a device that is used to measure the emf of an unknown cell. You can see a simple setup of a potentiometer in the figure given in the question.
We know that a cell creates a potential difference across a wire or a resistor. And due to resistance of the wire, there is a flow of charges and we say that there is a current in the circuit.
Therefore, there will be some current in the given circuit. A galvanometer is a device used to detect current in the circuit. Hence, if a current is flowing through the galvanometer, it will show some readings.
When the galvanometer shows no deflection or zero reading, there is no current flowing through the galvanometer. We say that the circuit is balanced and the length AJ is called balancing length. This is possible when the potential difference across the AJ is equal to the emf of the cell C.
Let the potential difference across AJ be V.
Hence, $V=\dfrac{E}{2}$
There will be a current in the loop consisting points A, B and D. Let that current be i. The total resistance in the loop is 12r+r = 13r. From Ohm’s law, we know V=iR. Hence, E=i(13r).
$\Rightarrow i=\dfrac{E}{13r}$.
Let the balancing length (AJ) be l and resistance of AJ be R’.
The potential difference across AJ is $V=\dfrac{E}{2}$. The current in AJ is $i=\dfrac{E}{13r}$.
Hence, V=iR’
$\Rightarrow \dfrac{E}{2}=\left( \dfrac{E}{13r} \right)R'$
$\Rightarrow \dfrac{1}{2}=\dfrac{R'}{13r}$
$\Rightarrow R'=\dfrac{13r}{2}$
The cross sectional area of wire AB be uniform and be A. Let its resistivity be $\rho $.
Therefore, $R'=\dfrac{\rho l}{A}$ ….. (i)
And
$12r=\dfrac{\rho L}{A}$ ….. (ii)
Divide equation (i) and equation (ii).
$\Rightarrow \dfrac{R'}{12r}=\dfrac{\dfrac{\rho l}{A}}{\dfrac{\rho L}{A}}$
$\Rightarrow \dfrac{R'}{12r}=\dfrac{l}{L}$
$\Rightarrow R'=\dfrac{12rl}{L}$
And we found that $R'=\dfrac{13r}{2}$
This means that $\dfrac{13r}{2}=\dfrac{12rl}{L}$
$\Rightarrow l=\dfrac{13L}{24}$
The balancing length is $\dfrac{13L}{24}$,
Hence, the correct option is D.
Note: Note that the emf of the unknown cell (i.e C) must be less than the emf of the main cell (D). Otherwise, we will not find a balancing length.
The emf of the unknown cell is given by $E=\dfrac{Vl}{L}$, where V is the potential difference across the resistance wire (AB).
The drawback of this method is that we cannot find the internal resistance of the unknown cell.
Recently Updated Pages
10 Examples of Evaporation in Daily Life with Explanations
10 Examples of Diffusion in Everyday Life
1 g of dry green algae absorb 47 times 10 3 moles of class 11 chemistry CBSE
What is the meaning of celestial class 10 social science CBSE
What causes groundwater depletion How can it be re class 10 chemistry CBSE
Under which different types can the following changes class 10 physics CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Who was the leader of the Bolshevik Party A Leon Trotsky class 9 social science CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Which is the largest saltwater lake in India A Chilika class 8 social science CBSE
Ghatikas during the period of Satavahanas were aHospitals class 6 social science CBSE