A diver at the bottom of a lake in Kashmir breathes out a bubble of air. At that depth the pressure is \(3\,\text{atm}\), and the bubble holds \(2\,\text{cm}^3\). The water has the same temperature everywhere. As the bubble rises, the pressure on it falls, and the bubble swells. At the surface, where the pressure is \(1\,\text{atm}\), it is much larger than when it left the diver.
The inverse law
Squeeze a gas slowly at a fixed temperature. Robert Boyle found that the pressure \(p\) and the volume \(V\) change in opposite ratios. Air at \(1\,\text{atm}\) in \(3\,\text{L}\) is at \(2\,\text{atm}\) in \(1.5\,\text{L}\) and at \(3\,\text{atm}\) in \(1\,\text{L}\). The product \(pV\) stays at \(3\,\text{atm L}\) throughout. A relation in which the product of two quantities stays fixed is an inverse law, \(y = k/x\).
A quantity \(y\) is inversely proportional to \(x\) when \(y = k/x\) for a constant \(k\). Then the product \(xy\) is the same for every pair of values. Multiplying \(x\) by any factor \(n\) divides \(y\) by \(n\). The graph of \(y = k/x\) is a hyperbola.
The figure shows what makes a hyperbola. Every point of the curve, joined to the axes, makes a rectangle of the same area \(k\). A wide rectangle must be short, and a narrow one must be tall. Other inverse laws of physics work the same way. At a fixed \(12\,\text{V}\), a \(4\,\Omega\) resistor carries \(3\,\text{A}\) and an \(8\,\Omega\) resistor carries \(1.5\,\text{A}\), because \(IR\) is fixed.
A hyperbola never reaches its axes
As \(x\) grows, \(k/x\) grows smaller and smaller, but it never becomes zero. As \(x\) shrinks towards zero, \(k/x\) grows without limit. So the curve comes closer and closer to both axes and never meets either of them. A line that a curve approaches in this way is called an asymptote. The two axes are the asymptotes of \(y = k/x\).
The asymptotes of a hyperbola need not be the axes. For a convex lens, the image distance \(v\) against the object distance \(u\) is a hyperbola. Its asymptotes are the lines \(u = -f\) and \(v = f\), where \(f\) is the focal length. An object far away gives an image close to the focus. An object brought towards the focus sends the image to a great distance.
A fixed amount of gas at a fixed temperature has its volume reduced to one quarter. What happens to its pressure?
The diver's bubble holds \(2\,\text{cm}^3\) at the bottom, where the pressure is \(3\,\text{atm}\). The temperature of the water is the same everywhere. What is the bubble's volume at the surface, at \(1\,\text{atm}\), in \(\text{cm}^3\)?
Turning a hyperbola into a straight line
A curved graph is hard to judge by eye. Is it really a hyperbola, or some other curve? Plot \(p\) against \(1/V\) instead of against \(V\). Since \(p = k \times (1/V)\), the new graph is a straight line through the origin, and its slope is \(k\). A laboratory tests Boyle's law this way, because a straight line is easy to recognise. The lesson on straight lines of this chapter explains slopes.
The inverse-square law
Some effects spread out from a point. The light of a lamp spreads over a sphere, whose area \(4\pi r^2\) grows as the square of the distance. So the intensity falls as \(1/r^2\). The force of gravity between two small bodies and the force between two charges also fall as \(1/r^2\). Such a relation, \(y = k/x^2\), is an inverse-square law.
For an inverse-square law, the ratio rule is \(y_2/y_1 = (x_1/x_2)^2\). Three times the distance gives one ninth of the force. The constant \(k\), such as \(Gm_1m_2\), cancels in the ratio and never needs to be known. This is why a ratio is the fastest way to use any of these laws.
Back to physics: the Moon falls like the mango
Newton asked whether the pull that brings a mango down also holds the Moon in its orbit. The mango is one Earth radius from the Earth's centre, and there the pull gives \(g = 9.8\,\text{m/s}^2\). The Moon is about \(60\) Earth radii from the centre. If the pull is an inverse-square law, it must be \(60^2 = 3600\) times weaker there.
At the Earth's surface, one Earth radius from its centre, the Earth's pull gives \(g = 9.8\,\text{m/s}^2\). The Moon is \(60\) Earth radii from the centre. If the pull follows an inverse-square law, what acceleration does it give the Moon?
The ratio rule of the inverse-square law is \(a_2/a_1 = (r_1/r_2)^2 = (1/60)^2 = 1/3600\).
So \(a_{\text{Moon}} = 9.8/3600 = 2.72 \times 10^{-3}\,\text{m/s}^2\).
The Moon's measured centripetal acceleration is \(0.0027\,\text{m/s}^2\). The inverse-square law passes the test, and the mango and the Moon obey one law.
Two small bodies attract with a force of \(3.6 \times 10^{-9}\,\text{N}\) when they are \(2\,\text{m}\) apart. They are moved to \(6\,\text{m}\) apart. Give the new force in units of \(10^{-10}\,\text{N}\).
A lamp gives an intensity of \(8\,\text{W/m}^2\) on a page \(2\,\text{m}\) away. The page is moved to \(4\,\text{m}\) away. What is the new intensity?
A student measures the pressure of a trapped gas at several volumes, at one temperature. Which graph should be a straight line through the origin?
An inverse law breaks down where its conditions fail. Boyle's law needs a fixed amount of gas at a fixed temperature, and a fast squeeze warms the gas. The inverse-square law of gravity holds outside a body. Inside the Earth, \(g\) falls in a straight line as you go deeper, to zero at the centre. And no real force becomes infinite: \(k/r^2\) has no value at \(r = 0\), where the bodies would overlap.
Boyle's law and the ideal gas equation in Kinetic Theory of Gases are inverse laws in \(p\) and \(V\). Newton's law of gravitation, the field of a point mass and the Moon test in Gravitational Force and Field are inverse-square laws. So are Coulomb's law, the intensity of a point source of sound or light, and the intensity of a radio source. Wien's law makes the peak wavelength inversely proportional to the temperature. A capacitor's reactance falls as the inverse of the frequency. The mirror and lens graphs of \(v\) against \(u\) are hyperbolas.
- Treating an inverse law as a subtraction.In \(y = k/x\) the product stays fixed. Doubling \(x\) halves \(y\); it does not lower \(y\) by a fixed amount.
- Using the square when the law is a plain inverse, or the reverse.Boyle's law has \(V\) to the first power; gravity and intensity have \(r^2\). Look at the power in the formula before you take a ratio.
- Expecting the hyperbola to reach an axis.\(k/x\) never becomes zero and has no value at \(x = 0\). The axes are asymptotes, approached but never met.