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:
32R
Understanding Electrical Resistance and Wire Dimensions
The resistance of a wire is a fundamental property that opposes the flow of electric current. It depends on the material of the wire, its length, and its cross-sectional area. For a cylindrical wire, the resistance (R) is directly proportional to its length (L) and inversely proportional to its cross-sectional area (A).
Formula for Resistance of a Wire
The formula for the resistance of a cylindrical wire made of a material with resistivity \(\rho\) is given by:
\[R = \rho \frac{L}{A}\]
Where:
- \(R\) is the resistance of the wire.
- \(\rho\) (rho) is the resistivity of the material. Resistivity is a property of the material itself and is constant for a given material at a specific temperature. Since both wires are made of the same material, \(\rho\) will be the same for both.
- \(L\) is the length of the wire.
- \(A\) is the cross-sectional area of the wire.
For a cylindrical wire with radius \(r\), the cross-sectional area is a circle, so its area is \(A = \pi r^2\). Substituting this into the resistance formula, we get:
\[R = \rho \frac{L}{\pi r^2}\]
Analyzing the Given Situation
We are given two cylindrical wires made of the same material. Let's denote the properties of the first wire with subscript 1 and the properties of the second wire with subscript 2.
Wire 1:
- Length, \(L_1 = L\)
- Radius, \(r_1 = r\)
- Resistance, \(R_1 = R\)
Using the resistance formula for Wire 1:
\[R = \rho \frac{L}{\pi r^2} \quad \text{(Equation 1)}\]
Wire 2:
The second wire is made of the same material (so resistivity \(\rho\) is the same). Its dimensions are related to the first wire's dimensions:
- Length, \(L_2 = \text{twice its length} = 2L\)
- Radius, \(r_2 = \text{one-fourth its radius} = \frac{r}{4}\)
We need to find the resistance of the second wire, \(R_2\). Using the resistance formula for Wire 2:
\[R_2 = \rho \frac{L_2}{\pi r_2^2}\]
Substitute the values \(L_2 = 2L\) and \(r_2 = \frac{r}{4}\) into the formula:
\[R_2 = \rho \frac{2L}{\pi \left(\frac{r}{4}\right)^2}\]
Calculating the Resistance of the Second Wire
Now, let's simplify the expression for \(R_2\):
\[R_2 = \rho \frac{2L}{\pi \left(\frac{r^2}{16}\right)}\]
To simplify the fraction, we can multiply the numerator by the reciprocal of the denominator:
\[R_2 = \rho \times 2L \times \frac{16}{\pi r^2}\]
Rearrange the terms to group \(\rho\), \(L\), and \(\pi r^2\) together:
\[R_2 = 32 \times \rho \frac{L}{\pi r^2}\]
From Equation 1, we know that \(R = \rho \frac{L}{\pi r^2}\). We can substitute \(R\) into the expression for \(R_2\):
\[R_2 = 32R\]
So, the resistance of the second wire is 32 times the resistance of the first wire.
Summary of Resistance Calculation
Let's summarize the dimensions and calculated resistance:
| Wire | Length (\(L\)) | Radius (\(r\)) | Cross-sectional Area (\(A = \pi r^2\)) | Resistance (\(R = \rho L/A\)) |
|---|---|---|---|---|
| Wire 1 | \(L\) | \(r\) | \(\pi r^2\) | \(R_1 = \rho \frac{L}{\pi r^2} = R\) |
| Wire 2 | \(2L\) | \(\frac{r}{4}\) | \(\pi \left(\frac{r}{4}\right)^2 = \pi \frac{r^2}{16}\) | \(R_2 = \rho \frac{2L}{\pi \frac{r^2}{16}} = \rho \frac{2L \times 16}{\pi r^2} = 32 \left(\rho \frac{L}{\pi r^2}\right) = 32R\) |
The resistance of the second wire is found to be \(32R\).
Revision Table: Factors Affecting Resistance
| Factor | Relationship with Resistance (assuming other factors constant) | Explanation |
|---|---|---|
| Length (\(L\)) | Directly proportional (\(R \propto L\)) | A longer wire offers more path for electrons to collide, increasing resistance. |
| Cross-sectional Area (\(A\)) | Inversely proportional (\(R \propto \frac{1}{A}\)) | A larger area provides more space for current flow, decreasing resistance. |
| Material (Resistivity, \(\rho\)) | Directly proportional (\(R \propto \rho\)) | Different materials have different inherent abilities to conduct electricity. High resistivity means high resistance. |
| Temperature | Generally, directly proportional for conductors | For most conductors, increased temperature leads to increased atomic vibrations, hindering electron flow and increasing resistance. |
Additional Information: Resistivity vs. Conductivity
Resistivity (\(\rho\)) is an intrinsic property of a material that quantifies how strongly it resists electric current. It is independent of the shape or size of the material.
The unit of resistivity is ohm-meter (\(\Omega \cdot m\)). Materials with low resistivity are good conductors (like copper, silver), while materials with high resistivity are poor conductors or insulators (like glass, rubber).
Conductivity (\(\sigma\)) is the reciprocal of resistivity, \(\sigma = \frac{1}{\rho}\). It measures how well a material conducts electric current. The unit of conductivity is siemens per meter (\(S/m\)). Good conductors have high conductivity, and insulators have low conductivity.
In this problem, since the material is the same for both wires, the resistivity \(\rho\) remains constant.
Similar Questions
If the length of a resistor is doubled, what happens to its resistance, assuming all other factors remain constant?
- The resistance of a particular material wire with length L and cross-sectional area A is 2 Ω. What would be the resistance of another wire of the same material having length 2L and area of cross-section A/2?
- Which of the following factors affect(s) the resistance of a wire?
- The resistance of a uniform metallic conductor depends on its:
- 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 same material but having four times its length and half its radius will be:
Important Questions from Resistance and Resistivity
Which of the following metals has the lowest electrical resistivity?
When electric current is passed through a wire, the amount of heat produced in a wire depends upon _______.
I. Length
II. Thickness
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 \(\rho = \frac{{Rl}}{A}\)
