Introduction to Electrostatics and Electric Charge

Introduction to Electrostatics

Have you ever noticed a small crackling sound while removing a sweater in dry weather? Or seen tiny pieces of paper getting attracted to a plastic comb after combing dry hair? These are common examples of electrostatic phenomena.

Electrostatics is the branch of physics where we study the behavior of electric charges at rest, the forces they exert, and the effects they produce. Think of it like the 'potential' energy of electricity before it starts flowing through wires as a current.

In daily life, electrostatic effects appear in:

• a spark while touching a metal doorknob,
• attraction of paper bits by a comb,
• lightning during thunderstorms,
• dust sticking to television or computer screens.

These effects arise because electric charge gets accumulated on bodies and then produces force or discharge.

Historically, the Greeks (Thales of Miletus) discovered that amber, when rubbed with wool, attracts light objects. In fact, the word 'electricity' comes from the Greek word elektron, which literally means amber.

Electrostatics deals with the study of forces, fields and potentials arising from static charges.

Electric Charge

Electric charge is a fundamental property of matter responsible for electric forces. Through simple experiments, like rubbing glass rods with silk, scientists realized there are exactly two types of charges. Benjamin Franklin named them:

• Positive charge
• Negative charge

The Golden Rule: Like charges repel each other ($+$ and $+$, or $-$ and $-$), and unlike charges attract each other ($+$ and $-$).

How do we detect charge?

A simple but effective device is the Gold Leaf Electroscope. It consists of a vertical metal rod in a jar with two thin gold leaves at the bottom. When a charged object touches the top, the charge spreads to the leaves. Since they get the same type of charge, they repel each other and spread apart!

Basic Properties of Electric Charge

Electric charge has certain fundamental properties that are very important for numerical problems as well as conceptual questions.

The three main properties are:

1. Additivity of charge
2. Conservation of charge
3. Quantisation of charge

1. Additivity of charge

If a system contains many charges, the total charge of the system is the algebraic sum of the individual charges.

If the charges are $q_1, q_2, q_3, \ldots, q_n$, then the net charge is: $$q_{\text{net}} = q_1 + q_2 + q_3 + \cdots + q_n$$

Why Algebraic Sum?

Because charge can be positive or negative.

So while adding charges, proper signs must be used.

For example, if a system has charges:

• $+2 \, \mu C$
• $-5 \, \mu C$
• $+4 \, \mu C$

then $$q_{\text{net}} = 2 - 5 + 4 = +1 \, \mu C$$

Key Point

Charge is a scalar quantity. It has magnitude and sign, but no direction.

This is why charges add like ordinary signed numbers, not like vectors.

2. Conservation of Charge

The total charge of an isolated system remains constant.

This means charge can neither be created nor destroyed in ordinary physical processes. It can only be transferred from one body to another.

Mathematical Statement

If an isolated system has total charge $Q$, then after any internal interaction, $$Q_{\text{initial}} = Q_{\text{final}}$$

Charging by Friction and Conservation

Suppose one neutral rod is rubbed with cloth.

• rod loses electrons and becomes positive,
• cloth gains electrons and becomes negative.

If the rod gets charge $+q$, the cloth gets charge $-q$.

So total charge remains: $$(+q) + (-q) = 0$$

Thus, no net charge is created.

Important Clarification

When we say charge is conserved, we mean net charge is conserved.

The distribution may change, but the total charge of the isolated system does not change.

Particle-Level Example

Sometimes particles may be created or destroyed in high-energy processes, but total charge still remains conserved.

For example, if a neutral neutron changes into a proton and an electron, then: $$0 = (+e) + (-e)$$

So total charge remains unchanged.

3. Quantisation of Charge

Electric charge exists in discrete packets.

The charge on any body is always an integral multiple of the basic unit of charge $e$.

Mathematically, $$q = n e$$ where:

• $n$ is an integer: $0, \pm 1, \pm 2, \pm 3, \ldots$
• $e$ is the magnitude of charge on one electron or proton

The value of elementary charge is: $$e = 1.6 \times 10^{-19} \, C$$

So possible charges are:

• $0$
• $+e$
• $-e$
• $+2e$
• $-3e$
• etc.

Meaning of Quantisation

A body cannot have charge like: $$2.5e$$ if we are speaking about free isolated charge in ordinary classical problems.

It must always be an integer multiple of $e$.

Why Macroscopic Charge Appears Continuous

For everyday objects, the number of excess or deficient electrons is extremely large. So the step size $e$ is too small to notice.

That is why in practical electrostatics, we often treat charge as if it were continuous.

For example, $$1 \, C = \frac{1}{1.6 \times 10^{-19}} \approx 6.25 \times 10^{18}$$ excess electrons in magnitude.

So one coulomb is actually a very large amount of charge.

Solved Examples

Example 1: The Basics of 1 Coulomb

How many electrons constitute 1 C of negative charge?

Solution:

1. Identify the values: Total charge $Q = 1$ C. Fundamental charge $e = 1.6 \times 10^{-19}$ C.
2. Use the formula: From $Q = ne$, we get $n = Q / e$.
3. Calculate: $n = 1 / (1.6 \times 10^{-19}) = 0.625 \times 10^{19}$.
4. Final Answer: There are $6.25 \times 10^{18}$ electrons in 1 C. This shows how huge a charge of 1 C actually is!

Example 2: Is this charge possible?

Can a body have a charge of $4.0 \times 10^{-19}$ C? Explain.

Solution:

1. Check for quantization: Calculate $n = Q / e$.
2. Substitute values: $n = (4.0 \times 10^{-19}) / (1.6 \times 10^{-19}) = 4.0 / 1.6 = 2.5$.
3. Analyze: Since $n$ must be an integer for a charge to exist independently, and 2.5 is not an integer, this charge is not possible.

Example 3: Net Charge Calculation

A system contains five charges: $+2 \mu C$, $+4 \mu C$, $-3 \mu C$, $-5 \mu C$, and $+1 \mu C$. What is the total charge?

Solution:

1. Apply Additivity: $Q = q_1 + q_2 + q_3 + q_4 + q_5$.
2. Sum them up: $Q = (2 + 4 - 3 - 5 + 1) \mu C$.
3. Result: $Q = -1 \mu C$. (Note: $1 \mu C = 10^{-6}$ C).

Example 4: Mass and Charge

If a neutral body gains $10^{12}$ electrons, what is its new charge? Does its mass change?

Solution:

1. Find Charge: $Q = -ne = -(10^{12} \times 1.6 \times 10^{-19}) = -1.6 \times 10^{-7}$ C.
2. Analyze Mass: Since electrons have mass ($m_e = 9.1 \times 10^{-31}$ kg), adding electrons increases the body's mass.
3. Calculate Change: $\Delta m = n \times m_e = 10^{12} \times 9.1 \times 10^{-31} = 9.1 \times 10^{-19}$ kg.
4. Conclusion: The mass increases by a very tiny, but real, amount.

Example 5: Alpha Particle Charge

An alpha particle is a helium nucleus ($He^{2+}$). What is its charge in Coulombs?

Solution:

1. Identify composition: An alpha particle has 2 protons and 0 electrons.
2. Use quantization: $Q = +ne = +2e$.
3. Calculate: $Q = 2 \times 1.6 \times 10^{-19} = 3.2 \times 10^{-19}$ C.

Example 6: Transfer Rate

A body emits $10^9$ electrons every second. How much time will it take to acquire a total charge of 1 C?

Solution:

1. Find charge per second: $rate = 10^9 \times 1.6 \times 10^{-19} = 1.6 \times 10^{-10}$ C/s.
2. Calculate time: $t = Q / rate = 1 / (1.6 \times 10^{-10}) = 6.25 \times 10^9$ seconds.
3. Convert to years: $6.25 \times 10^9 / (3600 \times 24 \times 365) \approx 198$ years. This example helps you visualize how large 1 C is compared to atomic transfers.

Example 7: Charge in a cup of water (JEE/NEET Standard)

Estimate the total positive charge in 250 g of water.

Solution:

1. Molar Mass of $H_2O$: $18$ g.
2. Find Molecules: $N = (Mass / Molar Mass) \times N_A = (250 / 18) \times 6.023 \times 10^{23}$.
3. Protons per molecule: Water ($H_2O$) has $2(1) + 8 = 10$ protons.
4. Total Protons: $n = N \times 10 = (250/18) \times 6.023 \times 10^{24}$.
5. Total Charge: $Q = ne = [(250/18) \times 6.023 \times 10^{24}] \times 1.6 \times 10^{-19} \approx 1.34 \times 10^7$ C.

Example 8: Sharing Charge

Two identical metal spheres A and B have charges $+10 \mu C$ and $-2 \mu C$. They are touched together and then separated. What is the final charge on each?

Solution:

1. Find Total Charge: $Q_{total} = +10 \mu C + (-2 \mu C) = +8 \mu C$.
2. Equal distribution: Since spheres are identical, the charge divides equally.
3. Final Result: $Q_A = Q_B = 8 / 2 = +4 \mu C$ each.

Example 9: Polythene Rubbing

A piece of polythene rubbed with wool is found to have a negative charge of $3.2 \times 10^{-7}$ C. Is there a transfer of mass? If so, how much?

Solution:

1. Determine n: $n = Q / e = (3.2 \times 10^{-7}) / (1.6 \times 10^{-19}) = 2 \times 10^{12}$ electrons.
2. Mass transfer: Since electrons move from wool to polythene, mass is transferred to polythene.
3. Calculate mass: $m = n \times m_e = 2 \times 10^{12} \times 9.1 \times 10^{-31} = 1.82 \times 10^{-18}$ kg.

Example 10: Conservation in Radioactivity

In the decay ${}^{238}_{92}U \rightarrow {}^{234}_{90}Th + {}^{4}_{2}He$, verify the conservation of charge.

Solution:

1. Initial Charge: $Q_i = 92e$.
2. Final Charge: $Q_f = 90e + 2e = 92e$.
3. Conclusion: Since $Q_i = Q_f$, the law of conservation of charge is satisfied.

Example 11: Induction Concept

A positively charged glass rod is brought near a neutral metallic sphere. The sphere is then grounded while the rod is still near. Finally, the ground is removed and then the rod is removed. What is the final state of the sphere?

Solution:

1. Step 1: The positive rod attracts electrons to the near side and repels positive centers to the far side of the sphere.
2. Step 2: Grounding allows electrons from Earth to flow to the sphere to neutralize the repelled positive side.
3. Step 3: Removing the ground traps the extra electrons on the sphere.
4. Step 4: Removing the rod allows the extra electrons to spread uniformly.
5. Result: The sphere becomes negatively charged.