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Q. 53

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
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Short Answer

The resistivity of a metal increases slightly with increased temperature. This can be expressed as , where T0 is a reference temperature, usually 20°C, and a is the temperature coefficient of resistivity.

a. First find an expression for the current I through a wire of length L, cross-section area A, and temperature T when connected across the terminals of an ideal battery with terminal voltage ∆V. Then, because the change in resistance is small, use the binomial approximation to simplify your expression. Your final expression should have the temperature coefficient a in the numerator.

b. For copper, a = 3.9 * 10-3 °C-1 . Suppose a 2.5-m-long, 0.40-mm-diameter copper wire is connected across the terminals of a 1.5 V ideal battery. What is the current in the wire at 20°C?

c. What is the rate, in A/°C, at which the current changes with temperature as the wire heats up?

a. We have found to be an expression for current.

b. The current in the wire at found to be .

c. The rate, in A/°C, at which the current changes with

temperature as the wire heats up found to be .

See the step by step solution

Step by Step Solution

Step 1: Given Information (Part a)

The resistivity of a metal increases slightly with increased temperature. This can be expressed as , where is a reference temperature, usually , and a is the temperature coefficient of resistivity

Step 2: Calculation(Part a)

We know that the resistance is given by

resistance, Length, Cross-section area , Temperature.

Substitute the given expression,

Gives us,

Appling Ohm's Law we have find intensity,

.

Step 3: Explanation(Part a)

Let us recall what the binomial approximation is:

Consider everything in the square brackets apart from the one as the in the above approximation

We can consider our expression in brackets,

As the is what puts it into the denominator. So, let us write the intensity again as

We can apply the binomial approximation,

We get,

.

Step 4: Given Information(Part b)

For copper, . Suppose a -long, -diameter copper wire is connected across the terminals of a ideal battery.

Step 5: Calculation(Part b)

We need to do before applying numerical values is substitute the expression for the area when the latter is a circle with the diameter known:

We will get that the intensity is

Substituting the numerical values,

We get,

The current in the wire at found to be

Step 6: Given Information(Part c)

The resistivity of a metal increases slightly with increased temperature. This can be expressed as , where is a reference temperature, usually , and a is the temperature coefficient of resistivity.

Step 9: Calculation(Part c)

What we are examining is the rate of change of the intensity with changing temperature; that is, the derivative with respect to the temperature of our expression for the intensity. This will be

Given numerical case, the answer will be,

Therefore, The rate, in A/°C, at which the current changes with temperature as the wire heats up found to be

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