Light, Electricity and Magnetism

Magnetic field and right-hand thumb rule

An electric current produces a magnetic field around the conductor that carries it, and the direction of this field is given by the Right-Hand Thumb Rule. Hold a straight conductor in the right hand with the thumb pointing along the current. The curled fingers then show the direction of the field lines, which form concentric circles around the wire in planes perpendicular to it. The rule can also be read backwards: if the field lines circulate anticlockwise as seen from one end, the current points toward the observer at that end.

Fleming's Left-Hand Rule is a different tool. It relates the force on a current-carrying conductor, the field and the current, with the thumb giving the force, the forefinger the field and the middle finger the current. It is therefore not the rule that gives the direction of the field around a wire.

The field at distance r from a long straight wire carrying current I is B = \dfrac{\mu_0 I}{2\pi r}. Outside the wire, B is inversely proportional to r at every distance, so doubling the distance halves the field.

A magnetic needle, which is a small freely pivoted magnet, aligns itself along a field and so reveals its presence and direction. An ammeter measures current and a voltmeter measures potential difference, so neither is a detector of a magnetic field.

A current-carrying solenoid behaves like a bar magnet, with one pole at each end. Inside a long solenoid the field is uniform, parallel to the axis, and given by B = \mu_0 n I, where n is the number of turns per unit length. The field therefore rises linearly with current.

Placing a soft iron bar inside the solenoid raises the permeability of the core, so the field grows and does not remain the same.

A charge q moving with velocity \vec v in a field \vec B experiences \vec F = q(\vec v \times \vec B), of magnitude F = qvB\sin\theta, where \theta is the angle between the velocity and the field. When the velocity is parallel or antiparallel to the field, \sin\theta = 0 and no force acts, so the charge keeps moving along its straight path.

Because the force depends on the sign of q, positive and negative charges moving together through a field perpendicular to their motion are pushed in opposite directions and separate. This separation is a direct consequence of the sign dependence of the Lorentz force.

Worked example: a 2 A current in a long straight wire gives at a distance of 0.1 m a field B = \dfrac{4\pi\times10^{-7}\times 2}{2\pi\times 0.1} = 4\times10^{-6}\ \mathrm{T}.

Mirror reflection and image formation

A plane mirror forms a virtual image. The reflected rays only appear to diverge from a point behind the mirror, so the image cannot be formed on a screen, although the eye sees it. The image is erect, has the same size as the object, is laterally inverted, and lies as far behind the mirror as the object lies in front of it. It is therefore never nearer to the mirror than the object is.

Under the New Cartesian Sign Convention, distances are measured from the pole of a mirror or the optical centre of a lens, with the object distance u taken as negative for a real object. The spherical mirror formula is \dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}. The lens formula carries a minus sign instead, \dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}, so the plus-sign version applies only to spherical mirrors.

A plane mirror has an infinite radius of curvature, so its focal length f = R/2 tends to infinity. In this limit the mirror formula gives v = -u, which is the plane mirror relation. The spherical formula can therefore be applied to a plane mirror only when the focal length goes to infinity.

For a spherical mirror the magnification is m = -\dfrac{v}{u}, and a segment perpendicular to the principal axis has its image length scaled by |m|. Segments at different distances from a concave mirror are magnified by different amounts. For example, an object at 2f has m = -1, while an object at 3f has v = -\dfrac{3f}{2} and |m| = \dfrac{1}{2}, so the image of the nearer segment is twice as long as that of the farther one.

For a person of height H, the smallest plane mirror that shows the full image is \dfrac{H}{2} high, and this length does not depend on how far the person stands from the mirror.

Worked example: a concave mirror with f = -10 cm and an object at u = -30 cm gives \dfrac{1}{v} = \dfrac{1}{f} - \dfrac{1}{u} = -\dfrac{1}{10} + \dfrac{1}{30} = -\dfrac{1}{15}, so v = -15 cm and m = -\dfrac{v}{u} = -\dfrac{1}{2}, a real, inverted image of half the size.

Electrical power formulas

Electrical power is the rate at which electrical energy is converted, and its SI unit is the watt. The basic definition is P = VI, where V is the potential difference in volts and I is the current in amperes. Ohm's law V = IR gives the equivalent forms P = I^2R and P = \dfrac{V^2}{R}, and all three are valid expressions for power.

A quick way to reject a wrong expression is to check its units. IR^2 has units of \mathrm{A\,\Omega^2}, which cannot reduce to watts, so it cannot represent power. Similarly, \dfrac{I^2}{R} has units of \mathrm{A^2/\Omega}, which is not the unit of power, so it is not a power expression.

The current drawn by a device from a supply is found from I = \dfrac{P}{V}. For example, a 100 W appliance on a 200 V supply draws I = \dfrac{100}{200} = 0.5 A.

Mind the unit prefixes before multiplying. Since 1\ \mathrm{mA} = 10^{-3}\ \mathrm{A}, a current of 250 mA must be written as 0.250 A, so a 200 V supply with this current delivers P = 200 \times 0.250 = 50 W.

The commercial unit of electrical energy is the kilowatt-hour, which is the energy used by a 1 kW load in one hour. Thus 1\ \mathrm{kWh} = 1000\ \mathrm{W} \times 3600\ \mathrm{s} = 3.6\times10^{6}\ \mathrm{J}.

For a fixed resistance, P \propto V^2, so halving the voltage reduces the power to one quarter of its value. For example, a lamp rated 200 V and 100 W has R = \dfrac{200^2}{100} = 400\ \Omega, and at 100 V it dissipates only \dfrac{100^2}{400} = 25 W.

Human eye defects and lenses

The human eye works like a camera in which the crystalline lens forms a real, inverted image on the retina. The eye focuses on objects at different distances by changing the curvature of its lens, a process called accommodation. The ciliary muscles contract to make the lens more convex for near objects and relax to flatten it for distant objects.

Because the focal length of the eye lens can vary, the eye lens is a lens of variable focal length. The iris changes the size of the pupil at the same time, which varies the aperture and the amount of light entering the eye. The eye therefore has a lens of variable focal length and a variable aperture.

In myopia, or short-sightedness, the eyeball is too long or the lens is too convergent, so the image of a distant object forms in front of the retina. Nearby objects remain clear, so a myopic person can read close print but cannot see distant objects distinctly.

Hypermetropia, or long-sightedness, is the reverse defect: the image of a nearby object is formed beyond the retina, so near objects look blurred while distant objects look clear. A convex lens corrects it. Presbyopia is the age-related loss of accommodation, which mainly affects near vision.

The power of a lens is P = \dfrac{1}{f} with f in metres. A convex lens has positive power and a concave lens has negative power. For example, a focal length of 0.25 m gives P = \dfrac{1}{0.25} = +4 D.

For myopia with a far point at distance x, the corrective concave lens must form a virtual image of a distant object at x. This requires f = -x and therefore P = -\dfrac{1}{x}. A far point of 4 m, for example, needs a lens of -0.25 D.

Refraction of light and speed in media

Light travels at about 3\times10^{8} m/s in air and vacuum, and its speed in any other medium is lower. The speed in a medium of refractive index n is v = \dfrac{c}{n}, so the speed depends on the medium.

When light passes from a denser medium into a rarer one, such as from water or glass into air, its speed increases. When it passes from air into a denser medium, it slows down. Any statement saying that light slows down on leaving water for air is therefore incorrect.

The frequency of a wave is set by its source and does not change at a boundary. The speed and wavelength change on entering a different medium, related by \lambda = \dfrac{v}{f}, but the frequency stays the same.

At normal incidence the angle of incidence is zero. Snell's law, n_1\sin i = n_2\sin r, then gives r = 0, so the ray passes straight into the denser medium without bending, although its speed decreases.

For a medium of refractive index n = 1.25, the speed of light is v = \dfrac{c}{1.25} = 0.8\,c. A refractive index greater than 1 always gives a speed less than c, because light cannot travel faster in a medium than in vacuum.

Refraction at a water-air surface makes an object under water appear closer to the surface than it really is. For viewing along the normal, the apparent depth is d_{\text{app}} = \dfrac{d}{n}, which is why the bottom of a tank looks raised. This effect is caused by refraction, not by reflection, interference or diffraction.

Atmospheric refraction phenomena

Twinkling of stars is caused by atmospheric refraction. Starlight travels through turbulent layers of air whose density and temperature vary continuously, and each layer changes the direction of the light. Because this variation is continuous and irregular, the apparent brightness of a star fluctuates, and this is seen as twinkling.

A star acts almost as a point source, so its fluctuating light is noticed as twinkling. A planet appears as a small disk and averages out these variations, so it does not twinkle noticeably. Phenomena such as diffraction, dispersion and polarization do not produce this effect.

A mirage forms when air near hot ground has successive layers of decreasing density. Light from the sky is refracted gradually as it passes through these layers, and near the ground it meets the layers at a large angle and undergoes total internal reflection toward the observer. A mirage is therefore an example of both refraction and total internal reflection.

A primary rainbow in water droplets forms through refraction on entry, one internal reflection inside the droplet, and refraction on exit. It is the brighter inner bow, with red on its outer edge and violet on its inner edge.

A secondary rainbow involves two internal reflections, so a primary rainbow cannot involve more than one. The primary bow also requires internal reflection, so refraction alone cannot produce it.

Electromagnetic induction and generators

Electromagnetic induction is the production of an electromotive force (emf) in a conductor when the magnetic flux linked with the conductor changes. Faraday's law states that the induced emf equals the rate of change of flux, \varepsilon = -\dfrac{d\Phi_B}{dt}, and the minus sign expresses Lenz's law, which says the induced current opposes the change that produces it. A generator uses this principle to convert mechanical energy into electrical energy, so it is the device that produces electric current, whereas a motor does the reverse by converting electrical energy into mechanical energy. Ammeters and galvanometers only measure current and do not produce it. For a coil of N turns and area A rotating at angular speed \omega in a uniform field B, the emf is \varepsilon = NBA\omega\sin\omega t. Fleming's right-hand rule gives the direction of the induced current for a conductor moving in a magnetic field: the forefinger points along the field, the thumb along the motion, and the stretched middle finger along the induced current. It is the generator counterpart of Fleming's left-hand rule, which gives the direction of the force on a current-carrying conductor in a motor. A DC generator works on Faraday's laws, because the changing flux through the rotating coil induces an emf, and Ohm's law and Joule's law do not explain this generation. The difference between an AC and a DC generator lies in the collector. A split-ring commutator reverses the connection to the external circuit every half rotation, so the output is unidirectional. Slip rings with brushes give an alternating output, and a stronger field or a different coil shape changes only the magnitude of the emf, not its direction.

Resistivity and resistance

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 therefore \rho = \dfrac{RA}{L}, and its SI unit is \Omega\,\mathrm{m}, obtained from \dfrac{\Omega\cdot\mathrm{m}^2}{\mathrm{m}}. The ohm is the unit of resistance, and ohm per metre is not the unit of resistivity. Resistivity is a property of the material and does not depend on the length or the area of cross-section, while resistance depends on all three. Both resistance and resistivity change with temperature, and for metals the resistivity rises as temperature rises. When a wire is stretched, its volume AL stays constant, so doubling the length halves the area and multiplies the resistance by four. For example, if R = 10\ \Omega and the length is doubled, the new resistance is 40\ \Omega, and at constant voltage the current falls to one-fourth. In a metal, the outermost electrons of each atom are free to move through the lattice. An applied electric field makes these free electrons drift, and they carry the current, whereas bound electrons stay attached to their atoms and the metal ions are fixed in the lattice.

Electromagnetic spectrum wavelengths

The electromagnetic spectrum is arranged by wavelength, which decreases in the order radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays. Approximate ranges are useful: visible light lies between about 400 nm and 700 nm, ultraviolet between about 10 nm and 400 nm, and X-rays between about 0.01 nm and 10 nm. Infrared lies roughly from 700 nm to 1 mm, and microwaves from about 1 mm to 1 m. Thus, microwaves have a longer wavelength than infrared, ultraviolet and X-rays, and a value of 1 nm identifies an X-ray. Since 1 Å equals 0.1 nm, a wavelength of 1 Å is also an X-ray wavelength. Infrared radiation is called heat radiation because matter absorbs it and converts it into thermal energy. Water molecules absorb infrared strongly through their vibrational and rotational modes, so this absorption explains why infrared heats water and moist objects effectively. X-rays are produced when high-energy electrons strike a metal target, and they are used in medicine, including radiotherapy for certain cancers. Radar systems use microwaves, which have much longer wavelengths, so X-rays are not used in radar.

Telescopes and optical instruments

A refracting telescope forms its image with lenses. The Galilean telescope uses a convex objective and a concave eyepiece, and the Keplerian telescope uses two convex lenses, so both are refractors. A reflecting telescope uses mirrors, and the Newtonian telescope is made only of mirrors: a concave primary mirror and a flat secondary mirror that directs the image to an eyepiece at the side of the tube. The Schmidt telescope combines a spherical mirror with a glass corrector plate, so it is not made only of mirrors. A compound microscope has an objective and an eyepiece, and both are convex lenses. The objective forms a real, magnified image and the eyepiece acts as a magnifier for it, so concave lenses are not used in this combination. Parts of a compound microscope include the mirror, which reflects light up through the specimen, the stage, on which the slide rests, and the stage clips, which hold the slide. The retina is a layer of the eye and is not a part of any microscope. In a telescope, the objective has the longer focal length and the larger aperture, so that it collects more light and forms a brighter image, while the eyepiece has the shorter focal length.