Physics 2004 Jamb Examination

To find the refractive index of the medium given the angles of incidence and refraction, we can use Snell's Law, which relates the angles of incidence and refraction to the refractive indices of the two mediums involved. Snell's Law is expressed as:

\[ n_1 \times \sin(\theta_1) = n_2 \times \sin(\theta_2) \]

Where:
- \( n_1 \) is the refractive index of the first medium (incident medium),
- \( n_2 \) is the refractive index of the second medium (refracted medium),
- \( \theta_1 \) is the angle of incidence,
- \( \theta_2 \) is the angle of refraction.

Given:
- \( \theta_1 = 60^\circ \)
- \( \theta_2 = 30^\circ \)

Assuming light travels from medium 1 (incident medium) to medium 2 (refracted medium), we have:

\[ n_1 \times \sin(60^\circ) = n_2 \times \sin(30^\circ) \]

We can rearrange this equation to solve for \( n_2 \):

\[ n_2 = \dfrac{n_1 \times \sin(60^\circ)}{\sin(30^\circ)} \]

Given that \( \sin(60^\circ) = \dfrac{\sqrt{3}}{2} \) and \( \sin(30^\circ) = 0.5 \), we can substitute these values:

\[ n_2 = \dfrac{n_1 \times \dfrac{\sqrt{3}}{2}}{0.5} \]

\[ n_2 = n_1 \times \sqrt{3} \]

So, the refractive index of the medium is \( \sqrt{3} \) times the refractive index of the incident medium.

Mirrors capable of producing parallel beams of light, such as those from the headlamps of a car, are typically parabolic mirrors. 

Parabolic mirrors

Parabolic mirrors are specifically designed to reflect incoming light rays parallel to the optical axis when the source of light is placed at the focal point of the mirror. This property makes them suitable for applications where parallel beams of light are required, such as in headlamp reflectors, satellite dishes, and telescopes.

A. Concave lens

Concave lenses diverge light rays, making them appear to come from a farther distance than they actually do. By using concave lenses.

C. Metal

The speed of sound depends on the medium through which it travels. In general, sound travels fastest through solids, followed by liquids, and then gases.

So, among the options provided:

• Metal: Sound travels fastest in solids, and metals are solid materials. Therefore, sound would travel fastest in metal.
• Water: Sound travels slower in liquids compared to solids, so it would be slower in water than in metal.
• Oil: Similar to water, sound travels slower in liquids, so it would also be slower in oil compared to metal.
• Gas: Sound travels slowest in gases among the three states of matter, so it would be slowest in a gas compared to the other options.

To calculate the angle of minimum deviation (\(D_{\text{min}}\)) for a ray passing through an equiangular prism, we can use the relation between the angle of minimum deviation (\(D_{\text{min}}\)) and the refractive index (\(n\)) of the prism:

\[ \sin\left(\dfrac{A+D_{\text{min}}}{2}\right) = n\sin\left(\dfrac{A}{2}\right) \]

Where:
- \( A \) is the angle of the prism (which is equal to the angle of incidence),
- \( n \) is the refractive index of the prism.

Given that the prism is equiangular, \( A = 60^\circ \), and the refractive index \( n = 1.4 \).

Let's calculate the angle of minimum deviation:

\[ \sin\left(\dfrac{60^\circ + D_{\text{min}}}{2}\right) = 1.4\sin\left(\dfrac{60^\circ}{2}\right) \]

\[ \sin\left(\dfrac{60^\circ + D_{\text{min}}}{2}\right) = 1.4\sin(30^\circ) \]

\[ \sin\left(\dfrac{60^\circ + D_{\text{min}}}{2}\right) = 1.4 \times \dfrac{1}{2} \]

\[ \sin\left(\dfrac{60^\circ + D_{\text{min}}}{2}\right) = 0.7 \]

To find \( \dfrac{60^\circ + D_{\text{min}}}{2} \), we need to find the angle whose sine is \( 0.7 \). Using inverse sine (sin⁻¹) function:

\[ \dfrac{60^\circ + D_{\text{min}}}{2} = \sin^{-1}(0.7) \]

\[ \dfrac{60^\circ + D_{\text{min}}}{2} \approx 44.43^\circ \]

Now, solving for \( D_{\text{min}} \):

\[ 60^\circ + D_{\text{min}} = 2 \times 44.43^\circ \]

\[ 60^\circ + D_{\text{min}} = 88.86^\circ \]

\[ D_{\text{min}} = 88.86^\circ - 60^\circ \]

\[ D_{\text{min}} = 28.86^\circ \]

Rounded to the nearest whole number, the angle of minimum deviation for the equiangular prism with a refractive index of 1.4 is approximately \(29^\circ\).

So, the correct answer is:
A. 29°

They meet at infinity

In a parallel beam of light, the rays travel in the same direction and do not converge or diverge significantly over short distances. 

C. \( \dfrac{v}{f} - 1 \)

The linear magnification (\(M\)) produced by a lens is given by the formula:

\[ M = \dfrac{\text{Image Height}}{\text{Object Height}} = \dfrac{v}{u} \]

Where:
- \( u \) is the object distance,
- \( v \) is the image distance.

For a convex lens, \( v \) is positive (real image) and \( u \) is positive (object on the same side as the incident light), so the linear magnification formula for a convex lens is:

\[ M = \dfrac{v}{u} \]

Now, let's find the expression that corresponds to this formula:

A. \( \dfrac{u}{v} + f \): This expression does not represent linear magnification, and it does not match the correct formula.
B. \( \dfrac{u}{f} - f \): This expression does not represent linear magnification and is not the correct formula.
C. \( \dfrac{v}{f} - 1 \): This expression represents linear magnification (\(M\)), where \(v\) is the image distance and \(f\) is the focal length. So, this expression matches the formula for linear magnification.
D. \( \dfrac{v}{f} + 1 \): This expression does not represent linear magnification and is not the correct formula

The correct answer is: B. 35°

If the ray of light makes an angle of 35° with a plane mirror, according to the law of reflection, the angle of reflection will also be 35°.

The pitch of a sound note depends on its frequency. 

D. Frequency

B. Be parallel

If the angle between two vectors P and Q is 0°, it means they are in the same direction or parallel to each other.