An electric circuit is consisting of a cell, an ammeter and a nichrome wire of length l. If the length of the wire is reduced to half (l/2), then the ammeter reading
gets doubled
Understanding Ammeter Reading Changes in an Electric Circuit
The question asks how the ammeter reading changes when the length of a nichrome wire in an electric circuit is reduced to half. An ammeter measures the electric current flowing through the circuit. The current in a circuit is determined by the voltage applied and the total resistance of the circuit.
Applying Ohm's Law
Ohm's Law states that the current (\(I\)) flowing through a conductor is directly proportional to the voltage (\(V\)) across its ends and inversely proportional to its resistance (\(R\)), provided the temperature remains constant. Mathematically, this is expressed as:
\(V = IR\)
or rearranging to find the current:
\(I = \frac{V}{R}\)
In this circuit, the cell provides a constant voltage (\(V\)). The ammeter has very low resistance, which can be considered negligible for this problem. The resistance comes mainly from the nichrome wire.
Resistance of the Nichrome Wire
The electrical resistance (\(R\)) of a wire depends on its material, length (\(l\)), and cross-sectional area (\(A\)). The formula for resistance is:
\(R = \rho \frac{l}{A}\)
where \(\rho\) (rho) is the resistivity of the material (nichrome in this case), which is a constant for a given temperature.
Analyzing the Change in Wire Length
Let's consider the initial and final states of the nichrome wire.
Initial State:
- Length of the wire = \(l\)
- Initial Resistance, \(R_1 = \rho \frac{l}{A}\)
- Initial Current, \(I_1 = \frac{V}{R_1}\)
Final State:
- Length of the wire = \(l/2\)
- The material (nichrome) and cross-sectional area (\(A\)) remain the same.
- New Resistance, \(R_2 = \rho \frac{l/2}{A} = \frac{1}{2} \left(\rho \frac{l}{A}\right)\)
- Comparing \(R_2\) with \(R_1\), we see that \(R_2 = \frac{1}{2} R_1\). The resistance is reduced to half when the length is reduced to half.
Calculating the New Ammeter Reading (Current)
The voltage (\(V\)) supplied by the cell remains constant. The new current (\(I_2\)) in the circuit with the reduced resistance is:
\(I_2 = \frac{V}{R_2}\)
Substitute the value of \(R_2\) in terms of \(R_1\):
\(I_2 = \frac{V}{R_1/2} = 2 \times \frac{V}{R_1}\)
Since \(I_1 = \frac{V}{R_1}\), we have:
\(I_2 = 2 \times I_1\)
This means the new current is twice the original current. The ammeter reading measures the current, so the ammeter reading will get doubled.
Summary of Changes
| Parameter | Initial Value | Change | Final Value |
|---|---|---|---|
| Wire Length (l) | \(l\) | Reduced to half | \(l/2\) |
| Resistance (R) | \(R_1 = \rho \frac{l}{A}\) | Reduced to half | \(R_2 = \frac{1}{2} R_1\) |
| Voltage (V) | \(V\) | Remains Constant | \(V\) |
| Current (I) | \(I_1 = \frac{V}{R_1}\) | Gets doubled | \(I_2 = \frac{V}{R_2} = 2I_1\) |
| Ammeter Reading | \(I_1\) | Gets doubled | \(I_2 = 2I_1\) |
Therefore, when the length of the nichrome wire is reduced to half, the resistance is also reduced to half. With constant voltage, the current in the circuit doubles. The ammeter reading, which measures this current, will consequently get doubled.
Revision Table: Electric Circuit Analysis
| Concept | Formula | Relationship with length | Relationship with Area |
|---|---|---|---|
| Resistance (R) | \(R = \rho \frac{l}{A}\) | Directly proportional (R \(\propto\) l) | Inversely proportional (R \(\propto\) 1/A) |
| Current (I) | \(I = \frac{V}{R}\) | Inversely proportional to R | Directly proportional to R |
Additional Information: Factors Affecting Resistance
The resistance of a metallic wire depends on the following factors:
- Length (\(l\)): Resistance is directly proportional to the length of the conductor. Longer the wire, more resistance it offers.
- Area of Cross-section (\(A\)): Resistance is inversely proportional to the area of cross-section of the conductor. Thicker wire means larger area, thus less resistance.
- Nature of the Material (\(\rho\)): Different materials have different resistivities. Conductors like copper and aluminum have low resistivity, while materials like nichrome or glass have high resistivity. Nichrome is often used in heating elements because of its high resistance.
- Temperature: The resistance of pure metals generally increases with an increase in temperature. For alloys like nichrome, the change in resistance with temperature is much less significant over a certain range compared to pure metals, but it still changes. In this specific problem, we assume temperature is constant unless stated otherwise.
An ammeter must always be connected in series in a circuit to measure the total current flowing through that part of the circuit.
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