Question 53: Current Electricity

Question

The current through a conductor is '$\mathrm{a}$' when the temperature is $0^{\circ} \mathrm{C}$. It is '$\mathrm{b}$' when the temperature is $100^{\circ} \mathrm{C}$. The current through the conductor at $220^{\circ} \mathrm{C}$ is

Accepted Answer

To find the current through the conductor at $220^{\circ} \mathrm{C}$, we need to understand the relationship between temperature and resistance, assuming that the resistance of the conductor increases linearly with temperature. This relationship is given by:
$ R_T = R_0 (1 + \alpha T) $
Where:
$ R_T $ is the resistance at temperature $ T $.
$ R_0 $ is the resistance at $0^{\circ} \mathrm{C}$.
$ \alpha $ is the temperature coefficient of resistance.
$ T $ is the temperature in degrees Celsius.
Since the current $ I $ is inversely proportional to the resistance $ R $ (Ohm's Law: $ I = \frac{V}{R} $), we'll first express the given currents $ a $ and $ b $ at the corresponding temperatures in terms of resistance:
At $0^{\circ} \mathrm{C}$:
$ I_0 = a = \frac{V}{R_0} $
At $100^{\circ} \mathrm{C}$:
$ I_{100} = b = \frac{V}{R_{100}} = \frac{V}{R_0 (1 + 100\alpha)} $
Dividing the equations for $ b $ by the equation for $ a $, we get:
$ \frac{b}{a} = \frac{V / R_{100}}{V / R_0} = \frac{R_0}{R_0 (1 + 100\alpha)} = \frac{1}{1 + 100\alpha} $
Solving for $ \alpha $, we get:
$ 1 + 100\alpha = \frac{a}{b} $
$ 100\alpha = \frac{a}{b} - 1 $
$ \alpha = \frac{1}{100} \left(\frac{a}{b} - 1\right) $
Now let's determine the current at $220^{\circ} \mathrm{C}$:
At $220^{\circ} \mathrm{C}$:
$ I_{220} = \frac{V}{R_{220}} = \frac{V}{R_0 (1 + 220\alpha)} $
Substituting the value of $ \alpha $ from above:
$ I_{220} = \frac{V}{R_0 \left(1 + 220 \cdot \frac{1}{100} \left(\frac{a}{b} - 1\right)\right)} $
$ I_{220} = \frac{V}{R_0 \left(1 + 2.2 \left(\frac{a}{b} - 1\right)\right)} $
$ I_{220} = \frac{V}{R_0 \left(1 + 2.2 \frac{a}{b} - 2.2\right)} $
$ I_{220} = \frac{V}{R_0 \left(\frac{b + 2.2 a - 2.2 b}{b}\right)} $
$ I_{220} = \frac{V b}{R_0 (b + 2.2 a - 2.2 b)} $
$ I_{220} = \frac{V b}{R_0 (2.2 a - 1.2 b)} $
Using $ I_0 = \frac{V}{R_0} $ from above:
$ I_{220} = \frac{b I_0}{2.2 a - 1.2 b} $
$ I_{220} = \frac{b a}{2.2 a - 1.2 b} $
Multiplying numerator and denominator by 5, we get:
$ I_{220} = \frac{5 a b}{11 a - 6 b} $
Therefore, the current through the conductor at $220^{\circ} \mathrm{C}$ is given by Option C:
$ \frac{5 a b}{11 a - 6 b} $

Answer

$\frac{5 a b}{11 b-6 a}$

Answer

$\frac{5 a b}{6 a-11 b}$

Answer

$\frac{5 a b}{11 a-6 b}$

Answer

$\frac{11 a b}{5 a-6 b}$

Current Electricity

The current through a conductor is '\mathrm{a}' when the temperature is 0^{\circ} \mathrm{C}. It is '\mathrm{b}' when the temperature is 100^{\circ} \mathrm{C}. The current through the conductor at 220^{\circ} \mathrm{C} is

Choose the correct answer: