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Question
How is potential gradient measured? Explain.
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Solution 1
Consider the following potentiometer, which is made up of a long uniform wire AB of length L and resistance R stretched on a wooden board and connected in series with a cell of stable emf E and internal resistance rand a plug key K.
Let I be the current flowing through the wire when the circuit is closed.
Current through AB, I = `"E"/("R" + "r")`
Potential difference across AB. VAB = IR
∴ `"V"_"AB" = "ER"/("R + r")`
The potential difference (the fall of potential from the high potential end) per unit length of the wire,
`"V"_"AB"/"L" = "ER"/(("R + r")"L")`
As long as E and r remain constant, `"V"_"AB"/"L"` will remain constant. `"V"_"AB"/"L"` is known as a potential gradient along with AB and is denoted by K. As a result, the potential gradient is computed by dividing the potential difference between the ends of the potentiometer wire by the wire's length.
Let P be any point on the wire between A and B and AP = l = length of the wire between A and P.
Then `"V"_"AP" = "Kl"`
∴ `"V"_"AP" prop l` as K is constant in a particular case. Thus, the potential difference across any length of the potentiometer wire is directly proportional to that length. This is the principle of the potentiometer.
Solution 2
- Potential gradient (K) is defined as a potential difference per unit length of wire.
- It is measured as, `"V"/"L" = "ER"/("L"("R" + "r"))`
where V = Potential difference between two points
L = Length (distance) between two points - Explanation:
- A potentiometer consists of a long wire AC of length L and resistance R having uniform cross-sectional area A.
- A cell of emf E having internal resistance r is connected across AC, as shown in the figure.
- When the circuit is switched on, the current I pass through the wire. Current through AC, I = `"E"/("R" + "r")` ….(1)
- Potential differences across AC,
VAC = IR
VAC = `"ER"/(("R" + "r"))` ….[From equation (1)] - Therefore, the potential difference per unit length of the wire is,
`"V"_"AC"/"L" = "ER"/("L"("R" + "r"))`
As long as E remains constant, `("V"_"AC")/"L"` will remain constant. - `("V"_"AC")/"L"` is known as a potential gradient along with AC and is denoted by K.
The potential gradient can be defined as a potential difference per unit length of wire.
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