The resistance (\(R\)) of a wire depends on its material (resistivity \(\rho\)), length (\(l\)), and cross-sectional area (\(A\)). The formula is:
Resistance Formula Basics
The relationship is given by:
\(R = \rho \frac{l}{A}\)
For a cylindrical wire with radius \(r\), the cross-sectional area is \(A = \pi r^2\). Substituting this, the resistance becomes:
\(R = \rho \frac{l}{\pi r^2}\)
Initial Wire Parameters
We are given:
- Length = \(L\)
- Radius = \(r\)
- Resistance = \(R\)
Using the formula, the initial resistance is:
\(R = \rho \frac{L}{\pi r^2}\)
Calculating New Resistance
For the second wire:
- Length (\(l_2\)) = \(4L\) (four times the original length)
- Radius (\(r_2\)) = \(r/2\) (half the original radius)
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 \(A_2\):
\(R_2 = \rho \frac{4L}{\frac{\pi r^2}{4}}\)
Simplify the expression:
\(R_2 = \rho \frac{4L \times 4}{\pi r^2} = \rho \frac{16L}{\pi r^2}\)
Recognize that \(\rho \frac{L}{\pi r^2}\) is the original resistance \(R\). Therefore:
\(R_2 = 16 \left( \rho \frac{L}{\pi r^2} \right) = 16R\)
The new resistance is \(16R\).
Similar Questions
- Which of the following statements is correct?
- 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 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 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?
