Physics: Electricity, Optics and Mechanics

Wave properties of sound

The speed of any wave is the product of its frequency and wavelength, v = f\lambda, and the time taken to cover a distance d is t = d/v. These two relations solve most sound problems, provided the speed used is the speed in the medium through which the sound actually travels.

Pitch is the perception of how high or low a sound is, and it is governed by frequency: a higher frequency gives a higher pitch. Loudness is governed by amplitude, while the speed of sound depends only on the medium and its temperature, not on pitch. Keep this mapping fixed: pitch corresponds to frequency, loudness to amplitude, and speed to the medium.

Sound in air is a longitudinal wave, because particles vibrate parallel to the direction of propagation and form compressions and rarefactions. Light is an electromagnetic wave whose oscillating electric and magnetic fields are perpendicular to the direction of travel, so it is transverse. Sound and light therefore belong to different wave types in air.

An echo is sound reflected back to the listener. Because the sound travels to the reflecting surface and back, the total path is twice the distance, so the distance to the surface is d = \dfrac{vt}{2}, where t is the time between the original sound and the echo. For example, an echo returning after 4 s in air, where the speed is 340 m/s, comes from a surface \dfrac{340 \times 4}{2} = 680 m away.

The speed of sound depends on the medium. It is highest in stiff, dense solids, where particles are tightly bound and pass vibrations on quickly. It is lower in liquids and lowest in gases. Among the media usually compared in examinations, stainless steel carries sound faster than glass, water or oxygen.

A sonic boom is a shock wave produced when a source moves faster than the speed of sound in the surrounding medium. The overlapping pressure waves merge into a cone-shaped front, which is heard as a loud boom. The ratio of source speed to sound speed is the Mach number, and a boom requires this ratio to exceed 1. A source moving at or below the speed of sound produces no shock front.

The wavelength is the length of one complete wave, so one crest together with one trough makes one wavelength. To find the wavelength from a count of waves, first find the distance travelled in the given time using d = vt, then divide by the number of complete wavelengths in that distance, \lambda = d/N.

Convex lens image formation

A convex (converging) lens forms images whose nature depends on where the object lies relative to the principal focus F and the point 2F. Using the sign convention in which distances measured against the incident light are negative, the lens formula is \dfrac{1}{v}-\dfrac{1}{u}=\dfrac{1}{f}, and the magnification is m = \dfrac{v}{u} = \dfrac{h'}{h}.

• Object beyond 2F — real, inverted and diminished image lying between F and 2F on the far side.
• Object at 2F — real, inverted image of the same size, formed at 2F on the other side. For f = 15 cm the object distance is 2f = 30 cm.
• Object between F and 2F — real, inverted and enlarged image formed beyond 2F. This is the principle of the projector.
• Object at F — refracted rays emerge parallel to each other, so the image forms at infinity. Substituting u = -f into the lens formula gives v = \infty.
• Object between F and the lens — virtual, erect and enlarged image on the same side as the object, which is the magnifier case.

The sign of the magnification tells the nature of the image. A positive magnification means the image is erect, and for a convex lens this means a virtual image. A magnification of +2 therefore describes an erect, virtual image twice the height of the object. Real images have negative magnification.

The aperture of a lens is the diameter of its effective circular boundary, set by its physical size and mounting. It is not the radius of curvature or the focal length. The curvature of the surfaces governs the focal length, whereas the aperture is a physical dimension of the lens, so it is independent of the radius of curvature.

A simple astronomical telescope uses two convergent lenses. The objective has a long focal length and forms a real, diminished image of a distant object, and the eyepiece has a short focal length and acts as a magnifier to view that image. Since the objective must form a real image, a divergent lens cannot be used as the objective.

For a quick check, an object placed 20 cm from a convex lens of focal length 10 cm gives \dfrac{1}{v} = \dfrac{1}{10} - \dfrac{1}{20} = \dfrac{1}{20}, so v = 20 cm and m = -1. The image is real, inverted and of the same size.

Electric power and current

Electrical power is the rate at which electrical energy is converted, given by P = VI. Combining this with Ohm's law gives the alternative forms P = I^2R = \dfrac{V^2}{R}. For a bulb operated at a potential difference V with current I, the power follows directly from these two readings.

The energy consumed is E = Pt. The commercial unit of energy is the kilowatt-hour, and one unit equals 3.6 \times 10^6 J. To convert, express power in kilowatts and time in hours. For example, a 60 W lamp used for 5 hours consumes 60 \times 5 = 300 Wh, that is 0.3 units.

A short circuit is a path of very low resistance connected across the source. By Ohm's law, I = \dfrac{V}{R}, so when R becomes very small the current rises sharply. This is why short circuits are dangerous and why fuses or circuit breakers are used to limit the current.

An ammeter is the instrument used to measure current. It is connected in series with the element so that the full current passes through it, and it is designed to have very low resistance so that it does not disturb the circuit. A voltmeter, by contrast, is connected in parallel and measures potential difference. Motors and generators are energy converters, not measuring devices.

In India, domestic supply is alternating current at 50 Hz, with a root mean square (r.m.s.) potential difference of 220 V between the live and neutral wires. The peak value is therefore 220\sqrt{2} \approx 311 V.

In a thermal power station, the chemical energy of fossil fuels such as coal, oil or natural gas is released as heat. This heat converts water into steam, and the steam drives a turbine coupled to a generator. The steam-turbine process is the mechanism by which the fuel produces electricity, so it explains the basic statement that thermal stations burn fuels to generate power.

Resistance and resistivity

The resistance of a uniform conductor is R = \rho\dfrac{L}{A}, where \rho is the resistivity, L the length and A the cross-sectional area. Resistivity is an intrinsic property of the material and depends on temperature, not on the size or shape of the piece. Two wires of the same metal at the same temperature therefore have the same resistivity even if their lengths and diameters differ.

Since A = \dfrac{\pi d^2}{4}, the resistance varies as R \propto \dfrac{1}{d^2} for a given metal and length. Doubling the diameter reduces the resistance to one-quarter of its original value, not one-half.

Resistance is directly proportional to length, R \propto L. If the length is halved while the source voltage remains the same, the resistance halves and the current I = V/R doubles, so an ammeter in the circuit shows a doubled reading.

Resistivity varies enormously across materials. Silver is an excellent conductor with resistivity of the order of 10^{-8}\ \Omega\,\text{m}, nichrome is an alloy with a much higher value of the order of 10^{-6}\ \Omega\,\text{m}, and glass is an insulator with a resistivity of 10^{10}\ \Omega\,\text{m} or more. The increasing order is silver, nichrome, glass.

For resistors in parallel, the reciprocals add: \dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3}. The equivalent resistance is always smaller than the smallest branch resistance. For example, resistors of 2 \Omega, 3 \Omega and 6 \Omega give \dfrac{1}{R} = \dfrac{3+2+1}{6} = 1, so R = 1\ \Omega.

Uniform circular motion

In uniform circular motion the speed is constant, but the velocity changes continuously in direction. The velocity is always tangent to the circular path. The acceleration is centripetal, a = \dfrac{v^2}{r}, and it points toward the centre. Because it is perpendicular to the velocity at every instant, it changes only the direction of motion and never the speed.

Displacement is the straight-line vector from the initial to the final position, whereas distance is the length of the path. After an exact whole number of revolutions the displacement is zero. After half a revolution the displacement equals the diameter, which is the largest possible chord of the circle.

The same reasoning applies to any fraction of a revolution. For instance, after three-quarters of a revolution on a circle of radius r, the displacement is \sqrt{2}\,r. When the number of revolutions is a half-integer, the body is diametrically opposite the starting point, and the displacement equals the diameter.

Every point on the Earth's surface completes one rotation in the same time, so the angular velocity \omega is the same everywhere. The linear velocity is v = \omega R\cos\phi, where \phi is the latitude. It is therefore greatest at the equator, where \phi = 0^\circ, and zero at the poles. Both the uniform angular velocity and the latitude-dependent linear velocity are correct.

When a stone is whirled on a string, the tension acts along the radius toward the centre, while the displacement at each instant is tangential. The angle between force and displacement is 90^\circ, so W = Fd\cos 90^\circ = 0. This holds for any mass, so the work done by the string is zero.

Refraction of light

Refraction is the bending of a light ray at the boundary between two transparent media. The frequency of light does not change at the boundary, but its speed does, because the refractive index of each medium differs. It is this change in speed that bends the ray; temperature and amplitude do not cause the bending.

When light passes from a rarer medium into a denser medium, its speed decreases and the ray bends towards the normal. Snell's law, n_1\sin\theta_1 = n_2\sin\theta_2, shows this directly: a larger refractive index in the second medium gives \theta_2 < \theta_1. The reverse case, from a denser to a rarer medium, speeds the light up and bends the ray away from the normal.

As a numerical check, a ray entering a medium of index 1.5 from air at 45^\circ has \sin\theta_2 = \dfrac{\sin 45^\circ}{1.5} \approx 0.47, so \theta_2 \approx 28^\circ, which is closer to the normal.

Twinkling of stars is caused by refraction in the atmosphere. Air layers at different temperatures and densities have different refractive indices, so starlight is bent continuously as it passes through them. The apparent brightness and position of the star fluctuate, which produces twinkling. Planets appear as small discs rather than point sources, so their fluctuations average out and they twinkle less.

The same atmospheric refraction explains why the flickering is attributed to refraction of starlight rather than to reflection, interference, diffraction or polarization. Each of those phenomena is a different effect and does not account for the rapid variation in apparent brightness.

Uniform acceleration and kinematics

Uniform acceleration means that the velocity changes by equal amounts in equal intervals of time, so the acceleration a is constant. For a body with initial velocity u that reaches final velocity v after time t, the first equation of motion is v = u + at, so a = \frac{v - u}{t}. Units must be consistent: a speed in km/h requires the time in hours, so a time given in minutes must be converted first. For example, a vehicle starting from rest that reaches 36 km/h in 30 minutes has a = \frac{36}{0.5} = 72\ \text{km/h}^2. Near the Earth's surface, a freely falling body experiences only gravity, so its acceleration stays constant at g \approx 9.8\ \text{m s}^{-2} downward. Its velocity grows as v = gt, so its momentum mv and kinetic energy \frac{1}{2}mv^2 increase while its gravitational potential energy mgh decreases. For a body decelerating uniformly to rest, the displacement is s = ut + \frac{1}{2}at^2, and the alternative form v^2 = u^2 + 2as gives the stopping distance s = \frac{u^2}{2|a|}. A scooter at 10 m/s decelerating at 5 m/s^2 therefore travels 10 m before stopping. On a velocity-time graph, velocity lies on the vertical axis and time on the horizontal axis, so the area between the graph and the time axis equals the net displacement, with areas below the time axis counted as negative. This area is not the total distance travelled, which needs the absolute area when velocity changes sign. Average speed is total distance divided by total time, not the arithmetic mean of the speeds unless equal times are spent at each speed. When equal distances are covered at speeds v_1 and v_2, the average speed is the harmonic mean \frac{2v_1v_2}{v_1 + v_2}.