Resistance

Linked From: 
The Liquid Nitrogen Show: the maths of low temperature physics

Problem set 1 : Resistance

In this set of problems we will learn about resistance and superconductivity. When a battery is connected to a circuit, current flows through the wires. It encounters a resistance when it does this, and this resistance depends on the temperature. If the temperature is very low then the resistance can drop to zero. We call this phenomenon superconductivity.

Consider the following circuit:

electric circuit.

Here we see a battery with a voltage V, connected in a circuit with a resistance, R. It generates a current, I. We will use a perfect battery for which V is always the same.

  1. Ohm's Law says that V, I and R are connected by the equation

    \begin{displaymath}V = IR \quad(1) \end{displaymath}

    where V is in volts, I is in amps, and R is in ohms. Draw a graph of I as a function of R. This means putting R on the horizontal axis, and I on the vertical one. Think about whether it makes sense to have negative values of R and I (remember what the symbols stand for, what the physical reality of each is). In a typical circuit you might make up in the Physics Lab I is about 1 amp, and R is about 10 ohms. What is the value of V? Choose values of R so that 10 ohms is roughly in the middle of your axis. How could you find the value of V directly from your graph?

  2. The value of R usually depends on the temperature, T, of the resistor (resistor is a name for something which has a resistance). Hotter things have a higher resistance, and it is known that there is a positive constant, K, so that

    \begin{displaymath}R = K(T + 273) \quad(2) \end{displaymath}

    where T is the temperature in degrees Celsius. Can you find out why the figure 273 is needed in this equation?
    Draw a graph of R as a function of T if K = 0.025. Using the formula (1) work out an expression for I in terms of T and draw a graph of this. What happens to I as T tends to absolute zero at -273C? This phenomenon is related to a property of some materials at low temperatures called superconductivity.

  3. A light bulb can take the place of the resistor. It has a resistance R and it is known that in a room at a temperature of 20C it has temperature, T, given by:

    \begin{displaymath}T = I^{2}R + 20 \quad(3) \end{displaymath}

    (In other words, the more current flowing through the bulb, the hotter it gets).
    Show that you can combine (1) and (3) so that:

    \begin{displaymath}T = \frac{V^{2}}{R} + 20 \end{displaymath}

    or

    \begin{displaymath}R = \frac{V^{2}}{(T-20)} \quad(4) \end{displaymath}

    Plot a graph of R as a function of T using formula (4) when V = 10volts. On the same graph plot R as a function of T using formula (2) with K = 0.025. From the point where these two graphs cross, can you work out the temperature of the light bulb?

    Harder: See if you can find the temperature, T, exactly in this case (this involves solving a quadratic equation). Can you find a formula for T which will work for any voltage, V?

  4. Find out as much as you can about superconductivity. Why might it be important for the future of technology?