Physical Sciences MCQ Quiz - Objective Question with Answer for Physical Sciences - Download Free PDF

Calculation:
We have ω_p as the angular frequency.

Thus its unit would be given as [T^-1]

We have the units of density (N) as 1/L³, charge (e) as IT, mass as M, and ε₀ is evaluated using the force equation F = 1 / (4πε₀) × (q₁q₂ / r²) as (I²T⁴) / (ML³).

By evaluating the units of each given option we get:

From dimension analysis: [T]^-1 = [N]^x[e]^y[M]^z[ε]^t ...(i)

Substituting the units: [T]^-1 = [1/L³]^x[IT]^y[M]^z[(I²T⁴) / (ML³)]^t.

Comparing the powers of M, L, I, and T:

z - t = 0 ...(ii)

-3x - 3t = 0 ...(iii)

y + 2t = 0 ...(iv)

y + 4t = -1 ...(v)

Solving equations: y = 1; t = -1/2; z = -1/2; x = 1/2.

Substituting x, y, z, and t in equation (i): ω_p = &sqrt;(N) e &sqrt;(m) &sqrt;(ε) = &sqrt;(Ne² / mε).

Since, A is orthogonal. Hence,

AA^T = I

⇒ \(\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & \rm a & \rm b \\ 0 & −\rm b & \rm a\end{array}\right)\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & \rm a & −\rm b \\ \rm a & \rm b & \rm a\end{array}\right) = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)\)

⇒ \(\left(\begin{array}{ccc} 1 & 0 & 0 \\\rm a b & \rm a^2+b^2 & 0 \\ \rm a & 0 & \rm a^2+b^2 \end{array}\right)=\left(\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)\)

⇒ a² + b² = 1 ⇒ a = \(\frac{1}{\sqrt{2}}\), b = \(−\frac{1}{\sqrt{2}}\)

CONCEPT:

Stefan's law: It states that the amount of radiation emitted per unit time per unit area of a black body at absolute temperature T is directly proportional to the fourth power of the temperature. 

Mathamatically, u = σT⁴

Where u = energy density, σ = Stefan's constant 

The power = P = energy density = energy density×area = u×A

EXPLANATION:

The absolute temperatures of A and B are T₁ = 400 K and T₂ = 200 K.

∴ The ratio of energy densities u₁/u₂ = (T₁/T₂)⁴ = (400/200)4 = 2⁴ = 16

∴ The power for the 1st body = P₁ = u×2A

The power of the 2nd body = P₂ = u×A

⇒ The power ratio = P₁/P₂ = (u₁×2A)/(u₂×A) = 2×16 = 32

Hence the correct answer is option 3.

CONCEPT:

The magnetic field at a distance 'r' from an infinitely long wire 

\(B =\frac{\mu_0}{4 \pi}\frac{2I}{r}\) (Where I is the current)

As shown in the figure the direction of B on the right side of the wire is inwards (⊗) and The direction of B on the left side of the wire is outwards (⊙)

EXPLANATION:

Let P be the point where the magnetic field becomes zero (as shown).

The distance from wires A, B, and C are (d-x), x, and (d+x)

So, the magnetic field due to A, B, and C

\(\begin{aligned} & ⇒B_A = \frac{\mu_0}{4 \pi} \frac{2I}{(d-x)} ⊗ \\ & ⇒B_B = \frac{\mu_0}{4 \pi} \frac{2I}{x} ⊙ \\ & ⇒B_C = \frac{\mu_0}{4 \pi} \frac{2I}{(d+x)}⊙ \\ \end{aligned}\)

∴ The total magnetic field at P 

\(\begin{aligned} &⇒ B_P = B_A + B_B+B_C \\ &⇒ B_P = \frac{\mu_0}{4 \pi} \frac{2I}{(d-x)} ⊗ + \frac{\mu_0}{4 \pi} \frac{2I}{x} ⊙ + \frac{\mu_0}{4 \pi} \frac{2I}{(d+x)}⊙ \\ & ⇒ B_P = \frac{2I\mu_0}{4 \pi} \left[ -\frac{1}{d-x} + \frac{1}{x}+ \frac{1}{d+x} \right] \\ & ⇒ B_P =\frac{2I\mu_0}{4 \pi} \left[ \frac{d^2 - 3x^2}{x(d-x)(d+x)} \right] \\ \end{aligned} \)

According to the problem, 

⇒ B_P = 0 (magnetic field vanishes)

\(\begin{aligned} & ⇒ \frac{2I\mu_0}{4 \pi} \left[ \frac{d^2 - 3x^2}{x(d-x)(d+x)} \right] = 0 \\ &∴ d^2 - 3x^2 = 0 \\ & x = \frac{d}{\sqrt{3}} \\ \end{aligned} \)

Like P there can be a similar point at \( x = -\frac{d}{\sqrt{3}}\) where also the magnetic field vanishes. 

So, the magnetic field vanishes at \( x = \pm \frac{d}{\sqrt{3}}\)

Hence the correct answer is option 4.

CONCEPT:

As we know;

Total energy = K.E +P.E

CALCULATION:

Given: K.E = \(\frac{GMm}{4a}\)

and Total energy, \( E= - \frac{{{\rm{GMm}}}}{{{\rm{2a}}}}\)

Now, P.E = \(\frac{GMm}{4a}-​​\frac{GMm}{2a}​​\)

⇒  P.E = \(-\frac{3GMm}{4a}\)

When a particle is the to distance from the center of the earth, then potential energy is written as;

P.E = \(-\frac{GMm}{r_o}\)

Now compare both we have;

\(-\frac{3GMm}{4a}=-\frac{GMm}{r_o}\)

⇒ \(r_o= \frac{4}{3} = 1.33\)

Hence option 1) is the correct answer.

Electromotive Force (EMF):

•  It is the force that causes to flow the free electrons in any closed circuit due to difference in electrical pressure or potential. 
• It is the voltage difference between the two terminals of a source in open circuit.
• It is the work done per unit charge by the source in taking the charge from lower to higher potential energy.

• Electromotive force is the electric potential generated by either an electrochemical cell or a changing magnetic field; It is also known as voltage.
• It is an electrical action produced by a non-electrical source, such as a battery (converts chemical energy to electrical energy) or generator (converts mechanical energy into electrical energy).
• Electromotive force is commonly denoted by the acronym emf, EMF or E.
• The SI unit for electromotive force is the volt.
• Electromotive force in a circuit maintains the potential difference.

\(EMF\; = \frac{{Energy\;in\;joule}}{{charge\;in\;coulombs}}\;\)

Dimensional Formula of Electromotive Force is ML2I-1T-3

Concept:

Capacitors in Series: When capacitors are connected in series, the equivalent capacitance (C_eq) is less than the smallest individual capacitance in the series combination.

The voltage across each capacitor can be different, but the charge (Q) on each capacitor is the same.

Equivalent Capacitance: \(\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \cdots + \frac{1}{C_n} \ \)

Voltage Distribution: The total voltage (V_total) across the series combination is the sum of the voltages across each capacitor:

\( V_{total} = V_1 + V_2 + V_3 + \cdots + V_n \ \)

Here, V_i is the voltage across capacitor (C_i).

Since the same charge (Q) is on each capacitor:

\(Q = C_1 V_1 = C_2 V_2 = C_3 V_3 = \cdots = C_n V_n \ \)

Capacitors in Parallel: When capacitors are connected in parallel, the total (equivalent) capacitance (C_{eq}) is the sum of the individual capacitances. The voltage (V) across each capacitor is the same, but the charge on each capacitor can be different.

Equivalent Capacitance: \(C_{eq} = C_1 + C_2 + C_3 + \cdots + C_n \ \)

Voltage Distribution: Each capacitor has the same voltage applied to it: \( V_{total} = V_1 = V_2 = V_3 = \cdots = V_n \ \)

Each capacitor stores a charge (Q_i) given by:  \(Q_i = C_i V \) 

The total stored charge (Q_total) is the sum of the individual charges:  

\(Q_{total} = Q_1 + Q_2 + Q_3 + \cdots + Q_n \ \)

Equivalent capacitance: \(\frac{1}{C_{eq}} = \sum \frac{1}{C_i}\ \)

Explanation:

From the given figure, it is clear that two capacitors are connected in parallel. So, the voltage across them is same.

d is also same for both capacitors.

Hence, the electric field is also the same by the following formula.

\({\rm{E}} = \frac{{{{\rm{ }}}{{\rm{ }}_{\rm{}}}{\rm{V}}}}{{\rm{d}}}\)

So, electrical filed in the region B is 4 kV/cm

Concept:

Right-Hand Screw Rule:

If the thumb is placed in the direction of current then the folding of the fingers will give the direction of the magnetic field, produced by the current-carrying conductor.

Faraday's Law of Electromagnetic Induction: 

When a conductor placed in a magnetic and if there is relative motion between them, then the conductor induces voltage within it. 

Lenz law: 

The direction of the electric current which is induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes the initial changing magnetic field.

Application:

• As the current is going in the upward direction and it will contain a magnetic field, so the direction of the magnetic field will be into the plane on the right side and out of the plane left side.
• Also, the current is decreasing in nature, so both side flux will be decreasing in nature.
• Due to interaction with the flux in the conducting loop, EMF is induced.

Loop A: The magnetic field in loop A is into the page. Since the flux is decreasing, the induced current in loop A will try to create more magnetic flux into the page. To achieve this, the current in loop A must flow clockwise.

Loop B: The magnetic field in loop B is out of the page. As the flux is decreasing, the induced current in loop B will try to create more magnetic flux out of the page. To achieve this, the current in loop B must flow counterclockwise.

Conclusion: The induced current in loop A will be clockwise and the induced current in loop B will be counterclockwise.

Dielectrics:

• A dielectric is an electrical insulator that can be polarised by an applied electric field.
• When a dielectric material is placed in an electric field, electric charges do not flow through because dielectrics do not have free electrons that may drift through the material instead they shift, from their average equilibrium positions, causing dielectric polarisation.
   

Explanation:

1. The dielectric materials have high resistivity.
2. The energy gap in the dielectric materials is very large. Hence, they have high breakdown voltage and used used as capacitors and insulators in electrical systems.
3. The temperature coefficient of resistance is negative and the insulation resistance is high. 
4. The attraction between the electrons and the parent nucleus is very strong. The electrical conductivity of these materials is very low as there are no free electrons to carry current.

These above properties shows that Dielectrics are like insulators and lack the physical structure and properties required which are required for superconductivity. Even at low temperatures, dielectrics do not exhibit superconducting properties.

Hence, they can not become superconductors as they are restricted by breakdown voltage and other factors.

Hence, the correct answer is option 1.

Concept

A diode conducts when the positive terminal of the battery is connected to the anode and the negative terminal of the battery is connected to the cathode. Under this condition, the diode is said to be in a forward-biased condition.

In forward-biased conditions, the diode is replaced by short-circuit.

A diode does not conduct when the positive terminal of the battery is connected to the cathode and the negative terminal of the battery is connected to the anode. Under this condition, the diode is said to be in a reverse-biased condition.

In reverse-biased conditions, the diode is replaced by an open circuit.

Note: When the battery of the same polarity is present on both sides of the diode, then the battery with greater voltage magnitude will decide the biasing of the diode.

Calculation

Case 1: During +ve half cycle:

Diode is forward-biased, hence the diode is replaced by short-circuit.

\(V_o=-V_s\)

Case 2: During -ve half cycle:

(i) 0< V_in < V

Diode is forward-biased, hence the diode is replaced by short-circuit.

\(V_o=-V_s\)

(ii) V < Vin < V_m

Diode is reverse-biased, hence the diode is replaced by an open-circuit.

\(V_o=0\)

So, the output waveform is:

So, the correct answer is option 2.

MMF:

We know that, Ohm’s law for magnetic circuit states that the MMF is directly proportional to the magnetic flux where reluctance is the constant of proportionality.

MMF = flux × reluctance

flux ∝ MMF  

↓ MMF → ↓ flux

Flux density (B): the amount of magnetic, electric, or other flux passing through a unit area.

\(B = \frac{ϕ }{A}\)

Flux: Magnetic flux is a measurement of the total magnetic field which passes through a given area, i.e. 

ϕ = B × A

Observation:

For a constant current around the coil, the magnetic field generated will also be constant, i.e. B = constant. Now for double the area, the net flux will be:

ϕ' = B × 2A

ϕ' = 2ϕ

Given that, Area (A) = 200 square centimetres

Number of turns (N) = 200

Change in magnetic field (ΔB) = 20 T

Change in time (Δt) = 1 second

The correct answer is Option 1.Key Points

• When an object undergoes acceleration, it means there is a change in its velocity. This change in velocity can occur either in terms of speed, direction, or both.
• A force always acts on it:
  • This statement is generally true.
  • According to Newton's second law of motion, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F = ma).
  • So, if there's acceleration, there must be a force acting on the object.

Additional Information

• Acceleration is a fundamental concept in physics that describes the rate of change of velocity with respect to time.
• Velocity is a vector quantity, meaning it has both magnitude (speed) and direction. Therefore, any change in speed, direction, or both constitutes acceleration.
• The formula for acceleration (a) is a = F/m where a is acceleration. F is the net force acting on an object and m is the mass of the object.
• This formula states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
• In simpler terms, if you apply a force to an object, it will accelerate, and the acceleration will be larger if the force is stronger or if the object has less mass.
• Acceleration can occur in various forms:
  • Linear Acceleration: Change in speed in a straight line.
  • Angular Acceleration: Change in rotational speed or direction.
  • Centripetal Acceleration: Acceleration directed towards the center of a circular path.
• Acceleration can be positive or negative:
  • Positive Acceleration: Speeding up in the positive direction.
  • Negative Acceleration (Deceleration): Slowing down or moving in the opposite direction.
• Gravity is a common force causing acceleration. Near the Earth's surface, objects in free fall experience acceleration due to gravity, denoted as g (approximately 9.8 m/s²).

Concept:

Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory.

Calculation:

v = \(\sqrt{2gh} \) and v = e\(\sqrt{2gh} \)

0 = (ev)² - 2gh₁

h₁ = \({e^2 × 2gh\over 2g}\) = e²h

Similarly, h₂ = e⁴h

H = h + 2h₁ + 2h₂ +...∞ 

= h + 2(e²h + e⁴h + ... ∞)

= h + 2e²h(\({1\over 1-e^2}\))

= h × (\({1+ e^2 \over 1-e^2}\))

The coefficient of restitution between the ball and the floor is 0.5.

e = 0.5

H = 5h/3

The correct answer is option (2).