Numerical for JEE:

1. A copper wire has a resistance of 10 ohms at 20°C. What will be its resistance at 50°C? (Coefficient of resistivity of copper = 0.0043/°C)

Solution:

Using the formula: $$R_t=R_0[1+\alpha (t_f-t_i)]$$

Where,

• (R_t) = Resistance at temperature (t_f)
• (R_0) = Resistance at temperature (t_i)
• (\alpha) = Coefficient of resistivity
• (t_f) = Final temperature
• (t_i) = Initial temperature

Substituting values: $$R_{50}=10[1+0.0043(50-20)]$$ $$R_{50}=10[1+0.0043(30)]$$ $$R_{50}=10[1+0.129]$$ $$R_{50}=10(1.129)$$ $$R_{50}=11.29ohms$$

Therefore, the resistance of the copper wire at 50°C is 11.66 ohms.

2. A germanium semiconductor has a resistivity of 0.4 ohm-cm at room temperature (27°C). What will be its resistivity at 100°C? (Coefficient of resistivity of germanium = -0.0048/°C)

Solution:

Using the formula: $${\rho }_t={\rho }_0[1+\alpha (t_f-t_i)]$$

Where,

• (\rho_t) = Resistivity at temperature (t_f)
• (\rho_0) = Resistivity at temperature (t_i)
• (\alpha) = Coefficient of resistivity
• (t_f) = Final temperature
• (t_i) = Initial temperature

Substituting values: $${\rho }_{100}=0.4[1-0.0048(100-27)]$$ $${\rho }_{100}=0.4[1-0.0048(73)]$$ $${\rho }_{100}=0.4[1-0.3504]$$ $${\rho }_{100}=0.4(0.6496)$$ $${\rho }_{100}=0.25984ohm-cm$$

Therefore, the resistivity of the germanium semiconductor at 100°C is 0.296 ohm-cm.

3. A nichrome wire has a resistance of 100 ohms at 25°C. What will be its resistance at 125°C? (Coefficient of resistivity of nichrome = 0.0004/°C)

Solution:

Using the formula: $$R_t=R_0[1+\alpha (t_f-t_i)]$$

Where,

• (R_t) = Resistance at temperature (t_f)
• (R_0) = Resistance at temperature (t_i)
• (\alpha) = Coefficient of resistivity
• (t_f) = Final temperature
• (t_i) = Initial temperature

Substituting values: $$R_{125}=100[1+0.0004(125-25)]$$ $$R_{125}=100[1+0.0004(100)]$$ $$R_{125}=100[1+0.04]$$ $$R_{125}=100(1.04)$$ $$R_{125}=104ohms$$

Therefore, the resistance of the nichrome wire at 125°C is 104.4 ohms.

Numerical for CBSE:

1. A carbon resistor has a resistance of 10 kΩ at 27°C. What will be its resistance at 77°C? (Coefficient of resistivity of carbon = -0.0005/°C)

Solution:

Using the formula: $$R_t=R_0[1+\alpha (t_f-t_i)]$$

Where,

• (R_t) = Resistance at temperature (t_f)
• (R_0) = Resistance at temperature (t_i)
• (\alpha) = Coefficient of resistivity
• (t_f) = Final temperature
• (t_i) = Initial temperature

Substituting values: $$R_{77}=10[1-0.0005(77-27)]$$ $$R_{77}=10[1-0.0005(50)]$$ $$R_{77}=10[1-0.025]$$ $$R_{77}=10(0.975)$$ $$R_{77}=9.75k\Omega$$

Therefore, the resistance of the carbon resistor at 77°C is 9.23 kΩ.

2. A nichrome wire is used as the heating element in a toaster. If the wire has a resistance of 12 ohms at room temperature (25°C), what will be its resistance when the toaster is operating at 250°C? (Coefficient of resistivity of nichrome = 0.0004/°C)

Solution:

Using the formula: $$R_t=R_0[1+\alpha (t_f-t_i)]$$

Where,

• (R_t) = Resistance at temperature (t_f)
• (R_0) = Resistance at temperature (t_i)
• (\alpha) = Coefficient of resistivity
• (t_f) = Final temperature
• (t_i) = Initial temperature

Substituting values: $$R_{250}=12[1+0.0004(250-25)]$$ $$R_{250}=12[1+0.0004(225)]$$ $$R_{250}=12[1+0.09]$$ $$R_{250}=12(1.09)$$ $$R_{250}=13.08ohms$$

Therefore, the resistance of the nichrome wire when the toaster is operating at 250°C is 16 ohms.