All practice questions

Question 13921Question →

In a standard Coolidge X-ray tube setup, the filament heating current is increased while maintaining a constant accelerating potential difference between the cathode and the target anode. Which of the following changes will be observed in the emitted X-ray beam?

The intensity of the X-ray beam increases while its minimum cut-off wavelength remains unchanged

The penetrating power of the X-ray beam increases while its intensity remains constant

The minimum cut-off wavelength of the X-ray beam decreases while its intensity increases

Both the maximum frequency and the overall intensity of the X-ray beam increase

Show answer & explanation

Answer: The intensity of the X-ray beam increases while its minimum cut-off wavelength remains unchanged

Answer

The intensity of the X-ray beam increases while its minimum cut-off wavelength remains unchanged

Increasing the filament current raises cathode temperature, causing thermionic emission of more electrons per second. This increases the total number of X-ray photons emitted per second, which means the intensity of the X-ray beam increases. Since the accelerating potential difference is held constant, the maximum kinetic energy imparted to each electron (

e V

) is unchanged. Consequently, the minimum cut-off wavelength given by

\lambda_{\min} = \frac{h c}{e V}

remains constant.

Step-by-Step Solution

1

Determine the physical effect of increasing the filament heating current

Higher filament current increases filament temperature, releasing a greater number of electrons per second via thermionic emission.

The rate of electron emission directly determines the number of X-ray photons generated per unit time, which defines the beam intensity.

2

Determine the physical effect of a constant accelerating potential difference

The maximum kinetic energy of striking electrons (

E_{\max} = e V

) and minimum wavelength (

\lambda_{\min} = \frac{h c}{e V}

) remain constant.

The accelerating voltage controls individual electron energy, which governs X-ray hardness and minimum cut-off wavelength via the Duane-Hunt law.

Key Concept

Intensity vs Hardness Control in X-ray Tubes

Estimated Time:1m 0s

Question 13922Question →

A photosensitive metal plate with a work function of $2.3\text{ eV}$ is illuminated by light of frequency $8.0 \times 10^{14}\text{ Hz}$ . If the intensity of the light source is doubled while maintaining the same frequency, what will be the new maximum kinetic energy of the emitted photoelectrons? ( $h = 6.6 \times 10^{-34}\text{ J s}$ , $1\text{ eV} = 1.6 \times 10^{-19}\text{ J}$ )

remain unchanged at

1.0\text{ eV}

double to

2.0\text{ eV}

quadruple to

4.0\text{ eV}

increase to

3.3\text{ eV}

Show answer & explanation

Answer: remain unchanged at $1.0\text{ eV}$

Answer

The maximum kinetic energy will remain unchanged at

1.0\text{ eV}

.

According to Einstein's photoelectric equation,

K_{\text{max}} = hf - W_0

. The maximum kinetic energy of an emitted photoelectron depends exclusively on the frequency of the incident photons and the work function of the target metal. Increasing the light intensity increases the rate of photon arrival and thus the rate of photoelectron emission, but it does not change the energy of individual photons. Therefore, the maximum kinetic energy remains constant at

1.0\text{ eV}

.

Step-by-Step Solution

1

Calculate the energy of the incident photons in Joules and convert to electron-volts (eV)

E = hf = 6.6 \times 10^{-34}\text{ J s} \times 8.0 \times 10^{14}\text{ Hz} = 5.28 \times 10^{-19}\text{ J}

. In eV:

E = \frac{5.28 \times 10^{-19}}{1.6 \times 10^{-19}} = 3.3\text{ eV}

.

Einstein's photoelectric equation requires comparing photon energy with the work function of the metal.

2

Calculate the maximum kinetic energy of the photoelectrons using Einstein's photoelectric equation

K_{\text{max}} = E - W_0 = 3.3\text{ eV} - 2.3\text{ eV} = 1.0\text{ eV}

.

The maximum kinetic energy is the surplus energy after overcoming the metal's work function.

3

Analyze the effect of doubling light intensity at constant frequency

Doubling intensity increases the photon flux (number of photons per second), thereby increasing emission current, but leaves individual photon energy and

K_{\text{max}}

completely unchanged at

1.0\text{ eV}

.

Kinetic energy of individual photoelectrons depends strictly on photon frequency, not beam intensity.

Key Concept

Independence of photoelectron kinetic energy from light intensity

Question 13923Question →

A spring balance suspended inside an elevator reads $48\text{ N}$ when supporting an object while the elevator accelerates downwards at $2\text{ m s}^{-2}$ . Taking acceleration due to gravity $g = 10\text{ m s}^{-2}$ , what is the mass of the object as measured by an equal-arm beam balance?

6.0\text{ kg}

4.0\text{ kg}

4.8\text{ kg}

60.0\text{ kg}

Show answer & explanation

Answer: $6.0\text{ kg}$

Answer

The mass of the object measured by an equal-arm beam balance is

6.0\text{ kg}

.

In a downward accelerating elevator, the apparent weight indicated by a spring balance is

W_{app} = m(g - a)

. Substituting

48\text{ N} = m(10\text{ m s}^{-2} - 2\text{ m s}^{-2}) = 8m

gives a mass of

6.0\text{ kg}

. Because a beam balance compares masses directly and both sides experience the exact same effective acceleration, it measures true mass regardless of the frame's acceleration.

Step-by-Step Solution

1

Formulate the equation for apparent weight in a downward accelerating reference frame.

W_{app} = m(g - a)

When an elevator accelerates downward, the effective acceleration experienced by an object inside is

(g - a)

.

2

Substitute the given values (

W_{app} = 48\text{ N}

,

g = 10\text{ m s}^{-2}

,

a = 2\text{ m s}^{-2}

) into the formula to find mass

m

.

48 = m(10 - 2) \implies 48 = 8m \implies m = 6.0\text{ kg}

This yields the true scalar mass of the object.

3

Determine the reading on an equal-arm beam balance.

The beam balance reads

6.0\text{ kg}

.

A beam balance compares unknown mass against standard counter-weights; both experience identical effective acceleration, making its measurement independent of motion or local gravity.

Key Concept

Apparent weight in an accelerating frame vs invariant mass measurement using a beam balance

Estimated Time:1m 0s

Question 13924Question →

Match each length measuring instrument on the left with its appropriate application and precision on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Metre rule pairs with measuring laboratory desk height (precision

1\text{ mm}

); Vernier caliper pairs with measuring internal tube diameter (precision

0.1\text{ mm}

); Micrometer screw gauge pairs with measuring glass cover slip thickness (precision

0.01\text{ mm}

); Measuring tape pairs with measuring distances over several metres.

Each instrument is correctly matched to its standard precision and intended application: the metre rule measures moderate lengths to within

1\text{ mm}

; the Vernier caliper measures internal and external dimensions to within

0.1\text{ mm}

; the micrometer screw gauge measures small thicknesses to within

0.01\text{ mm}

; and the measuring tape measures long or curved distances.

Step-by-Step Solution

1

Determine the least count (precision) and functional design of each measuring instrument.

Metre rule has a least count of

1\text{ mm}

; Vernier caliper has a least count of

0.1\text{ mm}

(

0.01\text{ cm}

) and possesses internal jaws; Micrometer screw gauge has a least count of

0.01\text{ mm}

; Measuring tape is flexible and suitable for long distances.

Each instrument is engineered for a specific range of dimensions and required degree of precision.

2

Match each instrument to the scenario requiring its specific precision and physical capability.

Metre rule matches desk height; Vernier caliper matches internal tube diameter; Micrometer screw gauge matches thin glass sheet thickness; Measuring tape matches large room dimensions.

Selecting the proper instrument minimizes measurement uncertainty and matches physical constraints.

Key Concept

Instrument selection based on least count, precision requirements, and physical geometry

Question 13925Question →

At a meteorological station in Maiduguri, Nigeria, the maximum shade air temperature recorded during a 24-hour period was $39.4^\circ\text{C}$ , while the minimum temperature recorded was $21.8^\circ\text{C}$ . What is the diurnal temperature range in degrees Celsius for that day?

Show answer & explanation

Answer: 17.6

Answer

The diurnal temperature range for the day is

17.6^\circ\text{C}

.

The diurnal range of temperature is defined as the difference between the highest (maximum) and lowest (minimum) temperatures recorded within a 24-hour period. Subtracting the minimum temperature (

21.8^\circ\text{C}

) from the maximum temperature (

39.4^\circ\text{C}

) yields

17.6^\circ\text{C}

.

Step-by-Step Solution

1

Identify the recorded daily maximum and minimum temperatures.

Maximum temperature =

39.4^\circ\text{C}

, Minimum temperature =

21.8^\circ\text{C}

.

Diurnal temperature range requires the highest and lowest values recorded during a 24-hour cycle.

2

Apply the diurnal temperature range formula.

Diurnal Range = Maximum Temperature - Minimum Temperature

The diurnal range represents the arithmetic difference between the maximum and minimum temperatures of the day.

3

Subtract the minimum temperature from the maximum temperature.

39.4^\circ\text{C} - 21.8^\circ\text{C} = 17.6^\circ\text{C}

Executing the subtraction gives the exact daily temperature variation.

Key Concept

Diurnal Temperature Range Calculation

Estimated Time:1m 0s

Question 13926Question →

In a photoelectric effect experiment using light of a frequency greater than the threshold frequency of a metal surface, doubling the intensity of the incident light doubles the maximum kinetic energy of the emitted photoelectrons.

True

False

Show answer & explanation

Answer: False

Answer

The statement is false. Doubling the light intensity increases the number of emitted photoelectrons per second but leaves their maximum kinetic energy unchanged.

The statement is false because the maximum kinetic energy of emitted photoelectrons is governed by Einstein's equation

K_{\text{max}} = hf - W_0

. It depends strictly on the frequency

f

of incident light and the metal work function

W_0

. Doubling the light intensity increases the rate of photon bombardment, which increases the number of photoelectrons emitted per unit time (photoelectric current), but leaves the maximum kinetic energy of individual photoelectrons completely unchanged.

Step-by-Step Solution

1

Identify the factors determining individual photon energy and maximum kinetic energy

Individual photon energy is given by

E = hf

, and maximum photoelectron kinetic energy is

K_{\text{max}} = hf - W_0

.

Einstein's photoelectric equation governs the energy exchange between a single incident photon and a single bound electron.

2

Analyze the physical meaning of light intensity in quantum terms

Intensity

I

is proportional to the number of photons striking the surface per unit time, not the energy of individual photons.

At a constant frequency

f

, changing intensity varies photon flux while individual photon energy

hf

stays constant.

3

Evaluate the effect of doubling light intensity on maximum kinetic energy

Doubling intensity doubles the rate of photoemission (photoelectric current) but does not change

K_{\text{max}}

.

Because

f

and

W_0

remain constant,

K_{\text{max}}

remains strictly unchanged.

Key Concept

Independence of photoelectron kinetic energy from light intensity

Estimated Time:1m 0s

Question 13927Question →

A pipe closed at one end vibrates in its fundamental mode with a frequency equal to that of an open pipe vibrating in its first overtone. What is the ratio of the length of the closed pipe to the length of the open pipe?

1 : 4

1 : 2

3 : 4

4 : 1

Show answer & explanation

Answer: $1 : 4$

Answer

The ratio of the length of the closed pipe to the length of the open pipe is

1 : 4

.

The fundamental mode of a pipe closed at one end has a frequency of

f = \frac{v}{4L_c}

. The first overtone of a pipe open at both ends corresponds to the second harmonic, which has a frequency of

f = \frac{v}{L_o}

. Equating the two frequencies gives

\frac{v}{4L_c} = \frac{v}{L_o}

, which simplifies directly to

\frac{L_c}{L_o} = \frac{1}{4}

or

1 : 4

.

Step-by-Step Solution

1

Express the fundamental frequency of the pipe closed at one end

f_c = \frac{v}{4L_c}

, where

v

is the speed of sound and

L_c

is the length of the closed pipe.

A pipe closed at one end supports odd harmonics, and its fundamental wavelength is

\lambda_c = 4L_c

.

2

Express the frequency of the first overtone of the open pipe

f_o = \frac{2v}{2L_o} = \frac{v}{L_o}

, where

L_o

is the length of the open pipe.

An open pipe supports all harmonics (

n = 1, 2, 3, \dots

). The fundamental is

n=1

and the first overtone corresponds to

n=2

.

3

Equate the two frequencies and solve for the length ratio

\frac{L_c}{L_o}

\frac{v}{4L_c} = \frac{v}{L_o} \implies 4L_c = L_o \implies \frac{L_c}{L_o} = \frac{1}{4}

The problem states that the fundamental frequency of the closed pipe is equal to the first overtone frequency of the open pipe.

Key Concept

Standing Waves and Harmonics in Pipes

Question 13928Question →

Match each physical quantity or measuring instrument related to mass and weight in Column A with its corresponding physical property or operational principle in Column B.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Beam balance matches with comparing gravitational forces using the principle of moments; Spring balance matches with measuring gravitational pull directly based on Hooke's law; Mass matches with an intrinsic scalar quantity measured in kilograms; Weight matches with a downward vector quantity measured in newtons.

Beam balances measure mass via the principle of moments independently of gravitational variation. Spring balances measure weight force using spring extension under Hooke's law. Mass is a constant scalar quantity measured in kilograms, while weight is a variable vector force measured in newtons.

Step-by-Step Solution

1

Identify the working principles of mass and weight measuring instruments.

The beam balance uses equal arms to balance moments (

m_1 g L = m_2 g L

), canceling

g

, so it measures invariant mass. The spring balance measures the force pulling a spring (

F = kx = mg

), depending directly on local

g

.

Instrument operation dictates whether mass or weight is measured.

2

Distinguish between the physical properties of mass and weight.

Mass is a scalar measure of quantity of matter (SI unit:

\text{kg}

) and is constant. Weight is the gravitational force acting on mass (SI unit:

\text{N}

) and is a vector quantity.

Fundamental physical definitions set the units and vector/scalar characteristics of mass versus weight.

Key Concept

Operating principles of balances and fundamental properties distinguishing mass from weight.

Estimated Time:1m 0s

Question 13929Question →

An electric water heater with an internal heating element of resistance $40\,\Omega$ is connected to a $200\,\text{V}$ mains power supply. If the heater is operated for $15\,\text{minutes}$ each day, what is the total electrical energy consumed by the heater over a period of $30\,\text{days}$ ?

7.5\,\text{kWh}

30.0\,\text{kWh}

450.0\,\text{kWh}

37.5\,\text{kWh}

Show answer & explanation

Answer: $7.5\,\text{kWh}$

Answer

The total electrical energy consumed over 30 days is

7.5\,\text{kWh}

.

The electrical power rating of the water heater is

P = \frac{V^2}{R} = \frac{200^2}{40} = 1000\,\text{W} = 1.0\,\text{kW}

. Operating for

15\,\text{minutes}

(

0.25\,\text{hours}

) daily for

30\,\text{days}

yields a total time of

7.5\,\text{hours}

. The total energy consumed is

1.0\,\text{kW} \times 7.5\,\text{h} = 7.5\,\text{kWh}

.

Step-by-Step Solution

1

Calculate the electric power rating of the heater

P = \frac{V^2}{R} = \frac{200^2}{40} = \frac{40000}{40} = 1000\,\text{W} = 1.0\,\text{kW}

Electric power dissipated in a resistance

R

connected across potential difference

V

is given by

P = \frac{V^2}{R}

.

2

Calculate the total operating time in hours

t = 30 \times \frac{15}{60} = 7.5\,\text{hours}

Commercial electrical energy is measured in kilowatt-hours (

\text{kWh}

), so time must be converted from minutes to hours.

3

Calculate total electrical energy consumed

E = P \times t = 1.0\,\text{kW} \times 7.5\,\text{h} = 7.5\,\text{kWh}

Electrical energy consumed is the product of power in kilowatts and total time in hours.

Key Concept

Calculation of commercial electrical energy consumption in kilowatt-hours using electric power and operating time.

Question 13930Question →

At a temperature of $27^\circ\text{C}$ , the root-mean-square (r.m.s.) speed of the molecules of an ideal gas is $300\text{ m/s}$ . What is the temperature of the gas, in degrees Celsius, when the r.m.s. speed of its molecules increases to $600\text{ m/s}$ ?

Show answer & explanation

Answer: 927

Answer

The temperature of the gas when the r.m.s. speed reaches

600\text{ m/s}

is

927^\circ\text{C}

.

According to kinetic theory, the root-mean-square speed of gas molecules is directly proportional to the square root of absolute temperature (

v_{\text{rms}} \propto \sqrt{T}

). First convert the initial temperature to Kelvin:

27^\circ\text{C} + 273 = 300\text{ K}

. Since the speed doubles from

300\text{ m/s}

to

600\text{ m/s}

, the ratio of speeds is

2

, which means the absolute temperature ratio is

2^2 = 4

. Thus, the new absolute temperature is

4 \times 300\text{ K} = 1200\text{ K}

. Converting back to Celsius gives

1200 - 273 = 927^\circ\text{C}

.

Step-by-Step Solution

1

Convert the initial temperature from Celsius to Kelvin

T_1 = 27^\circ\text{C} + 273 = 300\text{ K}

Gas kinetic equations require absolute temperature in Kelvin.

2

Apply the proportional relationship between r.m.s. speed and absolute temperature

\frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}}

In the kinetic theory of gases, root-mean-square speed is directly proportional to the square root of absolute temperature.

3

Calculate the final absolute temperature

T_2

T_2 = 1200\text{ K}

Doubling the r.m.s. speed requires quadrupling the absolute temperature (

2^2 \times 300\text{ K} = 1200\text{ K}

).

4

Convert the calculated absolute temperature back to degrees Celsius

\theta_2 = 1200 - 273 = 927^\circ\text{C}

Subtract 273 from the Kelvin temperature to find the value in degrees Celsius.

Key Concept

Proportionality between root-mean-square speed and absolute temperature in kinetic theory of gases

Question 13931Question →

A beam of monochromatic light strikes a photosensitive metal surface, causing the emission of photoelectrons. If the intensity of the incident light is doubled while maintaining the same frequency, what happens to the stopping potential of the emitted electrons?

It remains unchanged

It is doubled

It is quadrupled

It is reduced by half

Show answer & explanation

Answer: It remains unchanged

Answer

The stopping potential remains unchanged because it depends only on the frequency of the incident light, not its intensity.

According to Einstein's photoelectric theory, the maximum kinetic energy of emitted electrons is given by

K_{\text{max}} = hf - W_0

, where

h

is Planck's constant,

f

is the frequency of the incident light, and

W_0

is the work function of the metal. The stopping potential

V_s

is related to

K_{\text{max}}

by

e V_s = K_{\text{max}}

. Doubling the intensity of the light increases the number of photons hitting the surface per unit time (thus increasing the photocurrent), but it does not change the energy of individual photons (

hf

). Consequently, the maximum kinetic energy and the stopping potential remain completely unchanged.

Step-by-Step Solution

1

Recall Einstein's photoelectric equation

E_k^{\text{max}} = hf - W_0

The maximum kinetic energy of emitted photoelectrons depends directly on the photon energy (

hf

) and the work function (

W_0

) of the metal.

2

Relate maximum kinetic energy to stopping potential

e V_s = E_k^{\text{max}} = hf - W_0 \implies V_s = \frac{hf - W_0}{e}

Stopping potential (

V_s

) is directly proportional to the maximum kinetic energy.

3

Analyze the effect of changing light intensity at constant frequency

Increasing intensity increases the number of photons per second (and thus the photocurrent), but does not alter the energy of individual photons (

hf

). Therefore,

V_s

remains constant.

Intensity affects electron rate emission, not individual electron energy.

Key Concept

Independence of photoelectron kinetic energy and stopping potential from light intensity

Answer

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