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Ohm’s Law states that the current (I) through a conductor is directly proportional to the potential difference (V) across its ends, provided physical conditions like temperature remain constant: ( V = IR ). Example: If ( V = 12\,V ) and ( R = 4\,\Omega ), then ( I = 3\,A ).
Resistance ( R ) of a conductor is given by ( R = \frac{V}{I} ), measured in ohms (( \Omega )). It depends on material, length, area of cross-section, and temperature.
Resistivity ( \rho ) is an intrinsic property of a material: ( R = \rho \frac{l}{A} ), where ( l ) = length and ( A ) = cross-sectional area. Unit: ohm-meter (( \Omega\cdot m )).
Resistivity of metals increases with temperature due to increased lattice vibrations. For copper, resistivity at 20°C is approximately ( 1.7 \times 10^{-8}\, \Omega\cdot m ) (verify from NCERT).
Temperature dependence of resistivity for metals is given by: ( \rho_T = \rho_0 [1 + \alpha (T - T_0)] ), where ( \alpha ) is the temperature coefficient of resistivity.
Temperature coefficient of resistance ( \alpha ) is defined as: ( \alpha = \frac{R_T - R_0}{R_0 (T - T_0)} ), unit: ( ^\circ C^{-1} ).
For most metals, ( \alpha ) is positive (resistance increases with temperature); for semiconductors and insulators, ( \alpha ) is negative.
Example: Tungsten filament in bulb has high melting point and positive ( \alpha ); its resistance at operating temperature (~2500 K) is ~10 times higher than at room temperature.
Alloys like manganin and constantan have very low ( \alpha ), so their resistance remains nearly constant over a wide temperature range.
Superconductors have zero resistivity below a critical temperature ( T_c ). Example: Mercury becomes superconducting below 4.2 K (verify from NCERT).
Ohmic devices follow Ohm’s law (e.g., metallic conductors); non-ohmic devices do not (e.g., diode, transistor).
In a diode, V-I graph is non-linear; current flows only in forward bias, violating Ohm’s law.
Drift velocity ( v_d = \frac{eE\tau}{m} ), where ( \tau ) is relaxation time. It is directly related to current: ( I = n e A v_d ).
Electrical conductivity ( \sigma = \frac{1}{\rho} = \frac{n e^2 \tau}{m} ), where ( n ) = electron density.
Resistivity of semiconductors decreases with temperature because more charge carriers are excited into the conduction band.
For Nichrome, ( \alpha \approx 0.4 \times 10^{-3}\, ^\circ C^{-1} ); for copper, ( \alpha \approx 3.9 \times 10^{-3}\, ^\circ C^{-1} ) (verify from NCERT).
The relation between resistance and temperature: ( R_T = R_0 [1 + \alpha (T - T_0)] ) applies only to metals and within limited temperature ranges.
Resistivity is independent of the shape and size of the material; it depends only on the nature of the material and temperature.
Ohm’s law is not a universal law; it fails in non-ohmic materials like electrolytes, diodes, and gases.
Conductors obey Ohm’s law only when physical conditions (temperature, pressure, etc.) are constant.
Intermediate — Requires conceptual clarity between resistance and resistivity, and understanding of temperature effects on different materials.
Trap: Assuming all materials follow Ohm’s law. Avoid: Remember that Ohm’s law applies only to ohmic conductors (e.g., metals); diodes, transistors, and electrolytes are non-ohmic.
Trap: Confusing resistance with resistivity by thinking resistivity depends on dimensions. Avoid: Resistivity is material-specific and independent of length or area; resistance depends on both.
Trap: Applying ( R_T = R_0(1 + \alpha \Delta T) ) to semiconductors. Avoid: This formula is valid only for metals; semiconductors have decreasing resistance with temperature.
Q1. Which of the following materials has the highest resistivity? A) Copper B) Silver C) Manganin D) Silicon
Answer: D Explanation: Silicon is a semiconductor with higher resistivity than metals and alloys. Why others fail: Copper and silver are good conductors with very low resistivity; manganin has moderate resistivity but still much lower than silicon.
Q2. The resistance of a wire is ( R ). If its length is doubled and area of cross-section is halved, the new resistance becomes: A) ( R ) B) ( 2R ) C) ( 4R ) D) ( 8R )
Answer: C Explanation: Using ( R = \rho \frac{l}{A} ), new ( R' = \rho \frac{2l}{A/2} = 4 \rho \frac{l}{A} = 4R ). Why others fail: Students often forget that both length and area changes multiply the effect.
Q3. The temperature coefficient of resistance is negative for: A) Copper B) Aluminium C) Germanium D) Iron
Answer: C Explanation: Germanium is a semiconductor; its resistance decreases with temperature, so ( \alpha ) is negative. Why others fail: Metals like Cu, Al, Fe have positive ( \alpha ); students may assume all materials behave like metals.
Q4. A heating element is made of nichrome. Why is it preferred over copper? A) Nichrome has lower resistivity B) Nichrome has higher thermal conductivity C) Nichrome has high resistivity and high melting point D) Nichrome oxidizes easily
Answer: C Explanation: Nichrome has high resistivity and high melting point, making it suitable for heating elements. Why others fail: Students may think low resistivity is better, but high resistivity ensures more heat production.
Q5. At temperature ( T ), the resistance of a copper wire is ( R ). When cooled to ( T/2 ) (in Kelvin), its resistance: A) Increases B) Decreases C) Remains same D) Becomes zero
Answer: B Explanation: Copper is a metal; resistance decreases with decreasing temperature. Why others fail: Students may confuse with superconductors and think resistance becomes zero, but copper does not superconduct.
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