Given emf E = 2 V, internal resistance r = 0.5 ohm, external resistance R = 4.5 ohm. Current I = E/(R + r) = 2/(4.5 + 0.5) = 2/5 = 0.4 A. Terminal potential difference V = I R = 0.4 x 4.5 = 1.8 V (equivalently V = E - I r = 2 - 0.4 x 0.5 = 1.8 V).

I = E/(R + r); V = E - I r

The force on a current-carrying conductor is F = B I L sin theta. Here B = 0.2 T, I = 4 A, L = 0.5 m and theta = 90 deg (so sin theta = 1). F = 0.2 x 4 x 0.5 x 1 = 0.4 N. The direction of the force is given by Fleming's left-hand rule.

F = B I L sin theta

Consider a circular loop of radius R carrying current I. By the Biot-Savart law, each element I dl is perpendicular to the line joining it to the centre (theta = 90 deg), so dB = (mu0/4 pi)(I dl)/R^2. All elements produce fields in the same direction (perpendicular to the plane of the loop) at the centre, so they simply add. Integrating dl around the loop gives the circumference 2 pi R: B = (mu0/4 pi)(I/R^2)(2 pi R) = mu0 I/(2R). For N turns, B = mu0 N I/(2R).

B = mu0 I/(2R)

Ampere's circuital law states that the line integral of the magnetic field B around any closed loop is equal to mu0 times the total current enclosed by the loop: the integral of B.dl = mu0 I(enclosed). Applying it to a long solenoid with n turns per unit length carrying current I, the magnetic field inside is uniform and given by B = mu0 n I, directed along the axis; the field outside an ideal long solenoid is nearly zero.

B = mu0 n I

When a charged particle of charge q and mass m enters a magnetic field B perpendicularly with speed v, the magnetic force qvB provides the centripetal force m v^2/r. Equating: q v B = m v^2/r, so the radius r = m v/(q B). The time period T = 2 pi r/v = 2 pi m/(q B), which is independent of the speed and radius. This principle is used in the cyclotron.

r = mv/(qB); T = 2 pi m/(qB)

A moving coil galvanometer works on the principle that a current-carrying coil placed in a magnetic field experiences a torque. The deflection of the coil is directly proportional to the current passing through it (I is proportional to the deflection). To convert a galvanometer into an ammeter, a small resistance (shunt) is connected in parallel with it, so most of the current bypasses the galvanometer. To convert it into a voltmeter, a high resistance is connected in series with it, so that only a small current flows and it can measure potential difference.

The three elements of the earth's magnetism are: (1) Magnetic declination - the angle between the geographic meridian and the magnetic meridian at a place. (2) Magnetic dip or inclination - the angle made by the earth's total magnetic field with the horizontal at a place. (3) Horizontal component of the earth's field (BH) - the component of the earth's total magnetic field in the horizontal direction, given by BH = B cos(dip). These three completely specify the earth's field at a place.

BH = B cos(dip)

Consider a conducting rod of length l moving with velocity v perpendicular to a uniform magnetic field B. In time dt the rod sweeps an area dA = l (v dt), so the change in flux is d(flux) = B dA = B l v dt. By Faraday's law, the induced emf e = d(flux)/dt = B l v. Alternatively, the free electrons in the rod experience a force qvB that pushes them to one end, setting up an emf e = B v l across the rod.

e = B l v

Given N = 500, initial flux = 0.01 Wb, final flux = 0.05 Wb, so change in flux = 0.05 - 0.01 = 0.04 Wb, and time = 0.1 s. The induced emf (magnitude) = N (change in flux)/time = 500 x 0.04/0.1 = 500 x 0.4 = 200 V.

e = N (change in flux)/time

In a series LCR circuit connected to an AC source, the total opposition to current is called impedance Z, given by Z = sqrt(R^2 + (XL - XC)^2), where XL is the inductive reactance and XC is the capacitive reactance. The phase angle phi between the applied voltage and the current is given by tan(phi) = (XL - XC)/R. The current is I = V/Z.

Z = sqrt(R^2 + (XL - XC)^2)

Resonance in a series LCR circuit occurs when the inductive reactance equals the capacitive reactance (XL = XC). At resonance the impedance is minimum (Z = R), the current is maximum, and the circuit behaves as purely resistive. The condition XL = XC gives omega L = 1/(omega C), so omega^2 = 1/(LC), and the resonant frequency f = 1/(2 pi sqrt(LC)). Such a circuit is used for tuning radios and televisions.

f = 1/(2 pi sqrt(LC))

A transformer works on the principle of mutual induction: an alternating current in the primary coil produces a changing flux that induces an emf in the secondary coil. For an ideal transformer, Vs/Vp = Ns/Np = Ip/Is, where V is voltage, N is number of turns and I is current. A step-up transformer has more turns in the secondary (Ns > Np) and increases the voltage; a step-down transformer has fewer turns in the secondary (Ns < Np) and decreases the voltage.

Vs/Vp = Ns/Np = Ip/Is

For an ideal transformer, Ns/Np = Vs/Vp. Given Vp = 2200 V, Vs = 220 V, Np = 5000. So Ns = Np x (Vs/Vp) = 5000 x (220/2200) = 5000 x 0.1 = 500 turns. Since the secondary has fewer turns than the primary, it is a step-down transformer, as expected.

Ns/Np = Vs/Vp

In order of increasing frequency (decreasing wavelength): (1) Radio waves - used in radio and television communication. (2) Microwaves - used in radar and microwave ovens. (3) Infrared - used in remote controls and thermal imaging. (4) Visible light - used for vision and photography. (5) Ultraviolet - used to sterilise water and in detecting forgery. (6) X-rays - used in medical imaging and detecting fractures. (7) Gamma rays - used in cancer treatment (radiotherapy).

Using the sign convention, u = -20 cm and f = -15 cm. From the mirror formula 1/v = 1/f - 1/u = 1/(-15) - 1/(-20) = -1/15 + 1/20 = (-4 + 3)/60 = -1/60, so v = -60 cm. The magnification m = -v/u = -(-60)/(-20) = -3. The image is real (v negative), inverted (m negative) and magnified three times, formed 60 cm in front of the mirror.

1/v + 1/u = 1/f

Total internal reflection is the phenomenon in which a ray of light travelling from a denser to a rarer medium is completely reflected back into the denser medium when the angle of incidence exceeds a certain angle called the critical angle. Conditions: (1) light must travel from a denser to a rarer medium; (2) the angle of incidence must be greater than the critical angle C, where sin C = 1/n. Applications: optical fibres, sparkling of diamonds, and mirages.

sin C = 1/n

The critical angle C is given by sin C = 1/n, where n is the refractive index of the denser medium (glass). So sin C = 1/1.5 = 0.667, giving C = sin inverse (0.667) = 41.8 degrees (approximately 42 degrees). For angles of incidence greater than this, total internal reflection occurs.

sin C = 1/n

The lens maker's formula is 1/f = (n - 1)(1/R1 - 1/R2), where f is the focal length of the lens, n is the refractive index of the lens material with respect to the surrounding medium, and R1 and R2 are the radii of curvature of the two surfaces of the lens (taken with proper sign convention). It shows that the focal length depends on the material and the shape (curvature) of the lens.

1/f = (n - 1)(1/R1 - 1/R2)

Using the sign convention, u = -30 cm and f = +20 cm. From the lens formula 1/v - 1/u = 1/f, we get 1/v = 1/f + 1/u = 1/20 + 1/(-30) = 1/20 - 1/30 = (3 - 2)/60 = 1/60, so v = +60 cm. The magnification m = v/u = 60/(-30) = -2. The image is real (v positive for a lens), inverted (m negative) and magnified twice, formed 60 cm on the other side of the lens.

1/v - 1/u = 1/f

When a ray passes through a prism, it is deviated; the deviation is least at the angle of minimum deviation Dm, when the ray passes symmetrically through the prism. The refractive index of the prism material is given by n = sin((A + Dm)/2)/sin(A/2), where A is the refracting angle of the prism and Dm is the angle of minimum deviation. This formula is used to determine the refractive index of a transparent material.

n = sin((A + Dm)/2)/sin(A/2)