Physics 2004 Jamb Examination

The binding energy of a nucleus is the energy required to completely separate its constituent nucleons (protons and neutrons) from each other. It can be calculated using Einstein's mass-energy equivalence principle, \(E=mc^2\), where \(E\) is the binding energy, \(m\) is the mass defect, and \(c\) is the speed of light.

Given:
- Atomic mass of proton (\(m_p\)) = \(1.00783 \, \text{u}\)
- Atomic mass of neutron (\(m_n\)) = \(1.00867 \, \text{u}\)
- Mass number of helium (\(A\)) = \(4\)
- Atomic number of helium (\(Z\)) = \(2\)

The mass defect (\(\Delta m\)) can be calculated as the difference between the total mass of the nucleons in the helium nucleus and the actual mass of the helium nucleus:

\[ \Delta m = Z \cdot m_p + (A - Z) \cdot m_n - m_{He} \]

Where \( m_{He} \) is the atomic mass of helium.

\[ \Delta m = 2 \cdot 1.00783 \, \text{u} + (4 - 2) \cdot 1.00867 \, \text{u} - 4.0026 \, \text{u} \]

\[ \Delta m = 2.01566 \, \text{u} + 2 \cdot 1.00867 \, \text{u} - 4.0026 \, \text{u} \]

\[ \Delta m = 2.01566 \, \text{u} + 2.01734 \, \text{u} - 4.0026 \, \text{u} \]

\[ \Delta m = 4.033 \, \text{u} - 4.0026 \, \text{u} \]

\[ \Delta m = 0.0304 \, \text{u} \]

The binding energy (\(E\)) can be calculated using the mass-energy equivalence principle:

\[ E = \Delta m \cdot c^2 \]

\[ E = 0.0304 \, \text{u} \times (931.5 \, \text{MeV/u}) \]

\[ E \approx 28.34 \, \text{MeV} \]

Now, to convert MeV to unified atomic mass unit (u), we use the conversion factor:

\[ 1 \, \text{MeV} = \dfrac{1}{931.5} \, \text{u} \]

\[ 28.34 \, \text{MeV} = 28.34 \times \dfrac{1}{931.5} \, \text{u} \]

\[ E \approx 0.0304 \, \text{u} \]

Therefore, the binding energy of helium \(^4_2\text{He}\) is approximately \(0.0304 \, \text{u}\).

So, the correct answer is:
B. \(0.033 \, \text{U}\)

In a series \(RLC\) circuit at resonance, the inductive reactance \(X_L\) and the capacitive reactance \(X_C\) are equal in magnitude and cancel each other out. Mathematically, we have:

\[ X_L = X_C \]

So, the correct answer is:\[ \text{C. } X_L = X_C \]

Matter sometimes behaves as waves and other times as particles, therefore there are two theories of matter. Which are the wave theory and particle theory. 

The wavelength of a material object is given by:
$\lambda =\frac{h}{mv}$

Where, 

$\lambda$ = wavelength 
m = mass of the matter
v = velocity

Since momentum (P) = mv, the wavelength can be defined as:

$\lambda =\frac{h}{P}$

The negative temperature coefficient of resistance in semiconductors implies that their resistance decreases as temperature increases. This behavior is characteristic of semiconductor materials.

So, the correct answer is:
D. Their resistance decreases with temperature

Fluorescent tubes produce light through the excitation of gas molecules. When an electric current passes through the gas inside the tube, it excites the gas molecules, causing them to emit ultraviolet (UV) light. This UV light then interacts with the phosphor coating on the inside of the tube, causing it to fluoresce and emit visible light.

So, the correct answer is:
B. Excitation of gas molecules

In a reverse biased junction diode, the majority carriers (which are holes in p-type semiconductor and electrons in n-type semiconductor) are pushed away from the junction due to the applied reverse bias voltage. As a result, only minority carriers are available to contribute to the current flow.

So, the correct answer is:
C. Minority carriers

The energy stored in an inductor (\(W_L\)) is given by the formula:

\[ W_L = \dfrac{1}{2} L I^2 \]

Where:
- \( W_L \) is the energy stored in the inductor,
- \( L \) is the inductance of the inductor, and
- \( I \) is the current flowing through the inductor.

Given:
- Inductance (\( L \)) = \(5 \, \text{mH}\) = \(5 \times 10^{-3} \, \text{H}\)
- Current (\( I \)) = \(6 \, \text{A}\)

Substituting the values into the formula:

\[ W_L = \dfrac{1}{2} \times 5 \times 10^{-3} \times (6)^2 \]\[ W_L = \dfrac{1}{2} \times 5 \times 10^{-3} \times 36 \]\[ W_L = \dfrac{1}{2} \times 180 \times 10^{-3} \]\[ W_L = 90 \times 10^{-3} \]\[ W_L = 9.0 \times 10^{-2} \, \text{J} \]

Therefore, the energy stored in the inductor is \(9.0 \times 10^{-2} \, \text{J}\).

So, the correct answer is:
D. \(9.0 \times 10^{-2} \, \text{J}\)

X-rays are used in the study of crystal structures because they have an extremely short wavelength. This short wavelength allows X-rays to interact with the periodic structure of crystals, leading to diffraction patterns that provide information about the arrangement of atoms in the crystal lattice.

So, the correct answer is:
A. Have an extremely short wavelength

To find the total impedance (\(Z\)) of the circuit, we need to consider the impedance contributions from the resistor (\(R\)), inductor (\(X_L\)), and capacitor (\(X_C\)).

Given:
- Resistance (\(R\)) = \(8 \, \Omega\)
- Inductive reactance (\(X_L\)) = \(16 \, \Omega\)
- Capacitive reactance (\(X_C\)) = \(10 \, \Omega\)

The total impedance (\(Z\)) in a series \(RLC\) circuit is given by the formula:

\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]

Substituting the given values:

\[ Z = \sqrt{(8)^2 + (16 - 10)^2} \]\[ Z = \sqrt{64 + 36} \]\[ Z = \sqrt{100} \]\[ Z = 10 \, \Omega \]

Now, the total impedance is equal to the ratio of the voltage to the current in an AC circuit:

\[ Z = \dfrac{E}{I} \]

Where \(E\) is the emf (electromotive force) of the circuit and \(I\) is the current flowing through the circuit.

Given:
- Emf (\(E\)) = \(12 \, \text{V}\)

Substituting the values:

\[ 10 = \dfrac{12}{I} \]

Solving for \(I\):

\[ I = \dfrac{12}{10} = 1.2 \, \text{A} \]

Therefore, the current flowing in the circuit is \(1.2 \, \text{A}\).

So, the correct answer is:
C. \(1.2 \, \text{A}\)

To correct the error of producing a DC dynamo instead of an AC dynamo, the commutator should be replaced with slip rings. Slip rings are used in AC generators to allow the alternating current to be conducted out of the rotating armature coil without changing direction. In contrast, commutators are used in DC generators to convert the alternating current induced in the armature coil into direct current.

So, the correct answer is:
A. Commutators should be replaced with slip rings.