WAEC SSCE - Physics - 1996 (Essay)

(a) The apparent cubic expansivity of a liquid is the apparent increase in volume per unit original volume of the liquid per unit rise in temperature. It is the expansion observed when the expansion of the containing vessel is not allowed for.

(b)(i) Determination of apparent cubic expansivity

The apparatus is arranged as shown below.

A clean, dry density bottle fitted with a stopper having a fine capillary hole is weighed empty. Let its mass be \(M_1\). It is filled completely with the liquid at the initial temperature \(T_1\), the stopper is inserted, and the outside is wiped dry. The mass of the bottle and liquid is found as \(M_2\).

The bottle is suspended in a water bath, with the stopper and capillary orifice above the water level. A thermometer is placed in the bath. The water is heated slowly to a final temperature \(T_2\). As the liquid expands, some of it escapes through the capillary hole. Heating is continued until no further liquid escapes at \(T_2\).

The bottle is removed, allowed to cool, wiped dry and weighed again. Let its final mass be \(M_3\).

Thus,

Therefore, the apparent cubic expansivity is

(ii) Precautions

1. Heat the water bath slowly and stir it so that the liquid and the bath are at a uniform temperature.
2. Read the thermometer at eye level to avoid parallax error.
3. Keep the capillary orifice above the water level and wipe the bottle completely dry before each weighing.

(c) Calculation of the real cubic expansivity

For a temperature rise of \(50\,\mathrm{K}\):

The cubic expansivity of the glass bottle is

Since \(\gamma_r=\gamma_a+\gamma_g\),

(a) The statement means that the capacitor stores a charge of 2 microcoulombs for every volt of potential difference between its plates:

\[C=2\,\mu\text{F}=2\times10^{-6}\ \text{C V}^{-1}.\]

Thus, \(Q=CV\).

(b)(i) The capacitance of a parallel-plate capacitor depends on:

1. the overlapping area of the plates;
2. the separation between the plates;
3. the nature or permittivity of the dielectric material between the plates.

(b)(ii) Uses of capacitors include:

1. storing electric charge and energy;
2. smoothing the output of rectifier power supplies;
3. coupling alternating signals while blocking direct current in electronic circuits.

(c) When a capacitor is being charged, the potential difference is not constant. If the instantaneous charge on it is \(q\), then

\[V=\frac{q}{C}.\]

The work done in adding a small charge \(dq\) is

\[dW=Vdq=\frac{q}{C}dq.\]

Hence the energy stored when the charge increases from \(0\) to \(Q\) is

\[W=\int_0^Q\frac{q}{C}\,dq=\frac{1}{C}\left[\frac{q^2}{2}\right]_0^Q.\]

Therefore,

\[\boxed{W=\frac{Q^2}{2C}}\]

Since \(Q=CV\), equivalent forms are \(W=\frac12QV=\frac12CV^2\).

(d) The required parallel arrangement is shown below.

For capacitors in parallel, the potential difference across each capacitor equals the source voltage:

\[V_1=V_2=50\ \text{V}.\]

(i) Charge on either plate

For the \(2\,\mu\text{F}\) capacitor:

\[Q_1=C_1V=(2\times10^{-6})(50)=100\times10^{-6}\ \text{C}=\boxed{100\,\mu\text{C}}.\]

For the \(3\,\mu\text{F}\) capacitor:

\[Q_2=C_2V=(3\times10^{-6})(50)=150\times10^{-6}\ \text{C}=\boxed{150\,\mu\text{C}}.\]

Each capacitor has equal and opposite charges of these magnitudes on its two plates.

(ii) Potential difference across each capacitor

\[\boxed{V_1=V_2=50\ \text{V}}\]

(iii) Energy stored in the combination

\[C_{\text{total}}=C_1+C_2=2\,\mu\text{F}+3\,\mu\text{F}=5\times10^{-6}\ \text{F}.\]

\[W=\frac12C_{\text{total}}V^2=\frac12(5\times10^{-6})(50)^2.\]

\[\boxed{W=6.25\times10^{-3}\ \text{J}}\]

(a) Dispersion of white light is the separation of white light into its constituent colours when it passes through a refracting medium such as a glass prism.

(b) In ascending order of wavelength:

Violet, Indigo, Blue, Green, Yellow, Orange, Red (VIBGYOR).

(c)

1. The colour deviated least is red.
2. The colour deviated most is violet.

(d) White light is made up of colours of different wavelengths. The refractive index of glass is different for the different colours. Consequently, the colours travel at different speeds in the prism and are refracted through different angles. Violet light is refracted most, while red light is refracted least; hence the colours spread out to form a spectrum.

(e) Production of a pure spectrum of white light

An illuminated narrow slit is placed at the principal focus of a converging lens, L₁. The lens produces a parallel beam of white light which falls on a glass prism. The prism disperses the light into separate parallel beams of different colours. A second converging lens, L₂, brings each coloured beam to a separate focus on a white screen placed in its focal plane. Thus, sharp non-overlapping images of the slit form a pure spectrum.