Question 57: 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: