Resistance Calculation for Wire Dimensions Change
The resistance (\(R\)) of a wire depends on its material resistivity (\(\rho\)), length (\(L\)), and cross-sectional area (\(A\)). The formula is:
\( R = \rho \frac{L}{A} \)
For a cylindrical wire, the cross-sectional area is \(A = \pi r^2\), where \(r\) is the radius. Substituting this, the resistance becomes:
\( R = \rho \frac{L}{\pi r^2} \)
Initial Wire Resistance
Let the initial resistance be \(R_1\). Given:
- Length = \(L_1 = L\)
- Radius = \(r_1 = r\)
- Resistance = \(R_1 = R\)
So, the initial resistance is:
\( R = \rho \frac{L}{\pi r^1}^2 \)
New Wire Resistance Calculation
Let the new resistance be \(R_2\). The problem states the new wire has:
- Length = \(L_2 = L/2\)
- Radius = \(r_2 = r/2\)
- Material resistivity = \(\rho\) (same as initial)
First, calculate the new cross-sectional area (\(A_2\)):
\( A_2 = \pi r_2^2 = \pi \left(\frac{r}{2}\right)^2 = \pi \frac{r^2}{4} \)
Now, calculate the new resistance (\(R_2\)) using the resistance formula:
\( R_2 = \rho \frac{L_2}{A_2} \)
Substitute the values for \(L_2\) and \(A_2\):
\( R_2 = \rho \frac{L/2}{\pi r^2/4} \)
Simplify the expression:
\( R_2 = \rho \frac{L}{2} \times \frac{4}{\pi r^2} = \rho \frac{4L}{2\pi r^2} = \rho \frac{2L}{\pi r^2} \)
Relating New Resistance to Initial Resistance
We can rewrite \(R_2\) by factoring out the expression for the initial resistance (\(R\)):
\( R_2 = 2 \left( \rho \frac{L}{\pi r^2} \right) \)
Since \(R = \rho \frac{L}{\pi r^2}\), we have:
\( R_2 = 2R \)
Therefore, the resistance of the new wire is \(2R\).
Similar Questions
- Which of the following statements is correct?
- The length and the cross-sectional area of a conductor in the shape of a cylinder are '\(l\)' and '\(A\)' respectively. The resistivity of the material of the conductor is '\(\rho\)'. Its resistance depends upon
(p) \(l\) only
(q) \(l\) and \(A\)
(r) \(\rho\), \(l\) and \(A\)
(s) \(\rho\) and \(l\), but not \(A\)
Pick up the right option from those given above. - Resistivity of the wire depends on mainly ________
- The resistance of a metallic wire of length 1m and area of cross-section 1sq.metre having a resistivity is $2 \times 10^{-6}$ ohm - m is:
- A cylindrical wire of length \(L\) and radius \(r\) has a resistance \(R\). The resistance of another wire of same material but having four times its length and half its radius will be:
- A cylindrical wire of length L and radius r has a resistance R. The resistance of another wire of the same material but half its length and half its radius will be:
The resistance of a 10 m long copper wire is R ohm. What will be the resistance of a 5 m long copper wire?
When a number of resistance are connected in _______, their combined resistance is less than the smallest individual resistance.
A cylindrical wire of length L and radius r has resistance R. The resistance of another wire of the same material but of twice its length and one-fourth its radius is:
The resistance of a metallic wire is R. If the radius of the metallic wire is reduced to half of its initial value while keeping all other parameters the same, the new resistance becomes__________.
Important Questions from Resistance and Resistivity
Which of the following metals has the lowest electrical resistivity?
A uniform wire of resistance 9Ω is bent in the form of an equilateral triangle. Find the effective resistance across a side of the triangle.
The value of carbon resistance is 54 × 103 Ω. The percentage tolerance is 5%. What is the colour code sequence of carbon resistance?
Which of the following relations are wrong?
I. The specific conductance is given by the relation \(k = \frac{1}{R}\left( {l/A} \right)\)
II. The equivalent conducting is given by the relation \(\lambda = \frac{{100\;K}}{C}\)
III. The specific resistance is given by the relation \(\rho = \frac{{Rl}}{A}\)
A current is flowing through a metallic wire. If the wire is heated, which quantities change?
