Calculate the Temperature at Which the Resistance of a Metal Becomes 20% More Than Its Resistance at 300 K

Correct Answer: 1300 K

Understanding the Relationship Between Resistance and Temperature

The electrical resistance of a metal generally increases when its temperature increases. This happens because a rise in temperature causes the atoms of the metal lattice to vibrate more strongly. These increased lattice vibrations create greater opposition to the movement of free electrons, resulting in an increase in electrical resistance.

For a metal over the temperature range considered in this question, the variation of resistance with temperature can be expressed using the following relation:

R = R₀[1 + α(T − T₀)]

Here, R is the resistance at the final temperature T, R₀ is the resistance at the initial temperature T₀, α is the temperature coefficient of resistance, and (T − T₀) represents the change in temperature.

Given Information in the Question

The initial temperature of the metal is:

T₀ = 300 K

The temperature coefficient of resistance is:

α = 2.0 × 10⁻⁴ K⁻¹

The resistance becomes 20% more than its value at 300 K. We have to calculate the final temperature T at which this increase in resistance occurs.

What Does 20% More Resistance Mean?

Suppose the resistance of the metal at 300 K is R₀. An increase of 20% means that the resistance increases by 20% of its original value.

Increase in resistance = 20% of R₀

Increase in resistance = (20/100)R₀ = 0.20R₀

Therefore, the new resistance becomes:

R = R₀ + 0.20R₀

R = 1.20R₀

This conversion is important because the temperature-resistance equation requires the initial and final resistance values to be related mathematically.

Step-by-Step Calculation of the Required Temperature

Step 1: Apply the Resistance-Temperature Relation

The resistance of the metal at temperature T is given by:

R = R₀[1 + α(T − T₀)]

Since the initial temperature is 300 K, the equation becomes:

R = R₀[1 + α(T − 300)]

Step 2: Substitute the New Resistance

The resistance at the required temperature is 20% more than its resistance at 300 K. Therefore:

R = 1.20R₀

Substituting this value into the resistance-temperature equation gives:

1.20R₀ = R₀[1 + α(T − 300)]

Cancelling R₀ from both sides:

1.20 = 1 + α(T − 300)

Step 3: Substitute the Temperature Coefficient of Resistance

The given temperature coefficient is:

α = 2.0 × 10⁻⁴ K⁻¹

Therefore:

1.20 = 1 + (2.0 × 10⁻⁴)(T − 300)

Subtracting 1 from both sides:

0.20 = (2.0 × 10⁻⁴)(T − 300)

Step 4: Calculate the Change in Temperature

Rearranging the equation:

T − 300 = 0.20 / (2.0 × 10⁻⁴)

Simplifying:

T − 300 = 1000 K

Therefore:

T = 1000 + 300

T = 1300 K

Why Is the Required Temperature 1300 K?

The temperature coefficient of resistance of the metal is only 2.0 × 10⁻⁴ K⁻¹, which means that the fractional increase in resistance for each kelvin rise in temperature is relatively small. Therefore, a large temperature increase is required to produce a 20% increase in resistance.

The calculation shows that the temperature must rise by 1000 K above the initial temperature of 300 K. Hence, the final temperature is 1300 K.

Physical Meaning of the Temperature Coefficient of Resistance

The temperature coefficient of resistance indicates how strongly the resistance of a material changes with temperature. A positive value of α means that resistance increases as temperature increases, which is the typical behaviour of metals.

In this problem, α = 2.0 × 10⁻⁴ K⁻¹. This means that, according to the linear relation used in the question, the resistance changes by a fraction of 2.0 × 10⁻⁴ of its reference value for every 1 K increase in temperature.

Final Answer

The temperature at which the resistance of the metal becomes 20% more than its resistance at 300 K is 1300 K.

Using the relation R = R₀[1 + α(T − T₀)], the 20% increase in resistance corresponds to a temperature rise of 1000 K. Adding this increase to the initial temperature of 300 K gives the final temperature as 1300 K.