A silver sphere of radius 1 cm and work function 4.7 eV is suspended from an insulating thread in fr

Q: Why does the sphere stop emitting electrons eventually?

As electrons leave, the sphere becomes positively charged. This positive charge creates an electric potential that attracts the negatively charged electrons back. When this potential becomes equal to the stopping potential, even the most energetic electrons don't have enough kinetic energy to escape the sphere's pull, and net emission stops.

Q: Why do we use the value 1240 for 'hc'?

The value $1240 \text{ eV}\cdot\text{nm}$ is a very convenient approximation for the product of Planck's constant ($h$) and the speed of light ($c$). It allows you to quickly calculate photon energy in electron-volts (eV) when the wavelength is given in nanometers (nm), saving you from messy calculations with very small powers of 10.

Q: Can the sphere emit an infinite number of electrons if we keep the light on forever?

No. The emission is self-limiting. Once the sphere reaches the stopping potential, any newly ejected electron is immediately pulled back to the surface. The net number of electrons that have escaped remains constant.

Q: What would happen if we used light of a longer wavelength, say 300 nm?

A longer wavelength means lower photon energy. If the photon energy drops below the work function of silver ($4.7 \text{ eV}$), no electrons will be emitted at all, regardless of how intense the light is or how long you shine it.

Q: Why do we treat the sphere as a point charge when calculating its potential?

According to Gauss's Law, the electric field and potential outside a uniformly charged conducting sphere are mathematically identical to those of a point charge located at the center of the sphere. Therefore, we can use the standard point charge formula $V = kq/r$ at the surface.

Question

A silver sphere of radius 1 cm and work function 4.7 eV is suspended from an insulating thread in free-space. It is under continuous illumination of 200 nm wavelength light. As photoelectrons are emitted, the sphere gets charged and acquires a potential. The maximum number of photoelectrons emitted from the sphere is $A 	imes 10^Z$ (where $1 < A < 10$). The value of $Z$ is

Sigma Percentile · Accepted Answer

### The Setup: A Sphere in the Spotlight

Imagine a perfectly smooth silver sphere, suspended by an insulating thread in the vast emptiness of free space. We shine a continuous beam of ultraviolet light, with a wavelength of $200 	ext{ nm}$, directly onto it. What happens next is a beautiful interplay between quantum mechanics and classical electrostatics.

As the high-energy UV photons strike the silver surface, they transfer their energy to the electrons. If a photon's energy is greater than the work function of silver ($4.7 	ext{ eV}$), it knocks an electron completely out of the metal. This is the famous **photoelectric effect**.

### The Electrostatic Twist

But here is where the story gets interesting. Every time an electron escapes, it takes a negative charge with it. Because the sphere is isolated in space, it cannot replenish these lost electrons. Consequently, the silver sphere begins to acquire a net **positive charge**.

As more and more electrons are ejected, this positive charge grows. According to the laws of electrostatics, a positively charged sphere creates an electric potential around it. This potential acts like an invisible gravitational well, pulling the negatively charged escaping electrons back towards the sphere.

### The Tipping Point

Initially, the most energetic photoelectrons have enough kinetic energy to escape this potential well. However, as the sphere's positive charge increases, the potential well deepens. Eventually, a critical point is reached: the positive potential becomes so strong that even the fastest, most energetic photoelectrons are pulled back before they can escape to infinity.

At this exact moment, the net emission of electrons completely stops. The potential of the sphere has reached a value equal to the **stopping potential** ($V_0$) of the photoelectrons.

### The Master Equation

To find this stopping potential, we turn to Einstein's photoelectric equation:

$$K_{\max} = \frac{hc}{\lambda} - W = eV_0$$

We can calculate the energy of the incident photons using the handy shortcut $hc \approx 1240 	ext{ eV\cdot nm}$:

$$E_{	ext{photon}} = \frac{1240}{200} = 6.2 	ext{ eV}$$

Subtracting the work function of silver, we find the maximum kinetic energy of the ejected electrons:

$$K_{\max} = 6.2 	ext{ eV} - 4.7 	ext{ eV} = 1.5 	ext{ eV}$$

This means the stopping potential $V_0$ is exactly **$1.5 	ext{ V}$**.

### The Final Calculation

Now we know that the emission stops when the sphere's surface potential reaches $1.5 	ext{ V}$. From classical electrostatics, the potential $V$ of a conducting sphere of radius $r$ carrying a total charge $q$ is:

$$V = \frac{1}{4\pi\varepsilon_0} \frac{q}{r}$$

Since the total charge $q$ is simply the number of emitted electrons $n$ multiplied by the elementary charge $e$, we can write:

$$V_0 = \frac{1}{4\pi\varepsilon_0} \frac{ne}{r}$$

Let's plug in the values we know:
- $V_0 = 1.5 	ext{ V}$
- $\frac{1}{4\pi\varepsilon_0} = 9 	imes 10^9 	ext{ N\cdot m}^2/	ext{C}^2$
- $e = 1.6 	imes 10^{-19} 	ext{ C}$
- $r = 1 	ext{ cm} = 10^{-2} 	ext{ m}$

$$1.5 = \frac{(9 	imes 10^9) \cdot n \cdot (1.6 	imes 10^{-19})}{10^{-2}}$$

Simplifying the right side:

$$1.5 = \frac{14.4 	imes 10^{-10}}{10^{-2}} \cdot n$$

$$1.5 = 14.4 	imes 10^{-8} \cdot n$$

Solving for $n$:

$$n = \frac{1.5}{14.4 	imes 10^{-8}} = \frac{15}{144} 	imes 10^8 \approx 1.04 	imes 10^7$$

The problem states that the maximum number of photoelectrons emitted is $A 	imes 10^Z$. Comparing our result $1.04 	imes 10^7$ with this format, we can clearly see that $Z = 7$.

Enter Numerical Value:

Visualized Solution

$Initial Setup$

$Charging of the Sphere$

$Stopping Potential Condition$

$Substituting Values$

$Calculating V_{0}$

$Electrostatic Potential$

$Setting up the Equation$

$Solving for n$

$Final Conclusion$

The Sigma Insight: Photoelectric Effect

The Setup

The Electrostatic Twist

The Tipping Point

The Master Equation

The Final Calculation

Similar Questions

The maximum kinetic energy of photoelectrons emitted from a surface when photons of energy 6 eV fall on it is 4 eV. The stopping potential in volt is

The work function of a substance is $4.0 eV$ . The longest wavelength of light that can cause photoelectron emission from this substance is approximately

The work function of a substance is $4.0 eV$ . The longest wavelength of light that can cause photoelectron emission from this substance is approximately

Enter Numerical Value:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Initial Setup

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Charging of the Sphere

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Stopping Potential Condition

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Substituting Values

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Calculating V0​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Electrostatic Potential

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Setting up the Equation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Solving for n

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Conclusion

The Sigma Insight: Photoelectric Effect

The Setup

The Electrostatic Twist

The Tipping Point

The Master Equation

The Final Calculation

Similar Questions

The maximum kinetic energy of photoelectrons emitted from a surface when photons of energy 6 eV fall on it is 4 eV. The stopping potential in volt is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The work function of a substance is 4.0 eV. The longest wavelength of light that can cause photoelectron emission from this substance is approximately

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The work function of a substance is 4.0 eV. The longest wavelength of light that can cause photoelectron emission from this substance is approximately

$Initial Setup$

$Charging of the Sphere$

$Stopping Potential Condition$

$Substituting Values$

$Calculating V_{0}$

$Electrostatic Potential$

$Setting up the Equation$

$Solving for n$

$Final Conclusion$

The work function of a substance is $4.0 eV$ . The longest wavelength of light that can cause photoelectron emission from this substance is approximately

The work function of a substance is $4.0 eV$ . The longest wavelength of light that can cause photoelectron emission from this substance is approximately