capacitance and capacitors? Definition, formula, unit,

Introduction to Capacitors:

We all know that different objects can hold different amounts of electric charge, even when they are at the same voltage. This ability to store charge is called capacitance.

To store a useful amount of charge, we use a special device called a capacitor. A capacitor is made of two metal plates placed close to each other but separated by an insulating material like air, paper, or mica.

Think of a capacitor like a tiny rechargeable battery for electricity. It stores energy as an electric charge and can release it when needed. This makes capacitors very useful in many electrical circuits—from simple toys to big machines.

In direct current (DC) circuits, capacitors help control the flow of electricity in interesting ways. They are also used in radios, TVs, and other electronics to make sure everything works smoothly.

In this chapter, we’ll explore how capacitors behave in DC circuits and why they are so important. The ideas are simple, and you’ll see just how useful these little devices can be!

What is a Capacitor?

A capacitor is a device that stores electric charge.

It is made of two metal plates (or surfaces) that do not touch each other. Between these plates, there is a special material that does not let electricity pass through. This material is called a dielectric.

So, in simple words:

A capacitor has two metal plates, and between them is an insulating material.

The job of a capacitor is to hold and store electrical energy for a short time, and then give it out when needed.

Important Parts of a Capacitor:

1. Plates – These are the metal parts that hold the electric charge.
2. Dielectric – This is the insulating material between the plates. It can be air, mica, paper, plastic, ceramic, or something similar.

Things to Remember:

• The bigger the plates are, the more charge the capacitor can store.
• The closer the plates are to each other, the better the capacitor works.
• The type of dielectric also affects how much charge the capacitor can hold.

Different Types of Capacitors:

Capacitors are often named after the material used between their plates. For example:

• Air capacitor (uses air as dielectric)
• Paper capacitor (uses paper)
• Mica capacitor (uses mica)

Shapes of Capacitors:

Capacitors can come in different shapes like:

• Flat plates placed side by side
• Cylinders placed inside each other
• Or other special shapes, depending on where they are used

How Does a Capacitor Store Charge?

Let’s imagine a capacitor as a small box with two metal plates inside—plate A and plate B. These plates don’t touch each other. Between them, there is a special material that doesn’t let electricity pass through. This material is called an insulator or dielectric.

Now, what happens when we connect this capacitor to a battery?

1. Before Connecting the Battery
   When the capacitor is not connected to anything (the switch is open), both plates are neutral. That means they have no extra charge. They are just sitting there, doing nothing.
2. After Connecting the Battery
   When we close the switch and connect the battery, something interesting starts to happen.
   • The battery has two ends: a positive side (+) and a negative side (−).
   • The positive side of the battery pulls electrons (tiny particles with negative charge) away from plate A.
   • At the same time, the negative side pushes electrons onto plate B.

So now

• Plate A loses electrons and becomes positively charged.
• Plate B gains electrons and becomes negatively charged.

This is what we call charging a capacitor. It means the capacitor is now storing energy in the form of electric charge on its plates.

3. When Charging Stops
   The charging keeps going until the voltage (push) from the battery is equal to the voltage across the capacitor. Once both are equal, no more electrons move, and the charging stops. At this point, the capacitor is fully charged.

Why Is It Called a Capacitor?
It’s called a capacitor because it has the capacity to store electric charge. Earlier, it was also called a condenser because it “condenses” or packs electric field lines into a small space between its plates.

How a Capacitor Works

A capacitor is a small electrical device that can store electric charge. Let’s understand how it works in a simple way.

Imagine a capacitor like a sandwich. It has two metal plates with a gap in between. This gap is filled with air or a special material that doesn’t let electricity pass through.

Now, here’s what happens when you connect a battery to it:

Charging the Capacitor
When the battery is connected, it pushes electrons (tiny particles that carry electricity) from one plate to the other.

One plate starts to lose electrons and becomes positively charged.

The other plate gains electrons and becomes negatively charged.
This whole process is very quick and takes just a tiny moment.

Current Stops Flowing
After some time, the plates can’t hold any more charge. At that point, the current stops flowing. This means the capacitor is now fully charged.

No Flow Between Plates
It’s important to know that electricity does not pass directly between the two plates of the capacitor. Instead, electrons just move from one plate to the other through the wires outside.

Equal and Opposite Charges
The two plates always have the same amount of charge, but in opposite directions.

If one plate has +5 units, the other will have –5 units.
That’s why we say a capacitor has a certain amount of charge – it just means each plate holds that much, but in opposite forms.

Who Gives the Energy?
The energy needed to move these electrons and charge the plates comes from the battery. It works hard to transfer the electrons from one side to the other.

What Happens When the Switch is Opened?

If you open the switch (disconnect the battery), the charges stay on the plates. They don’t go anywhere. This is how we know that a capacitor can store charge and hold it for some time.

Capacitance

Capacitance is the ability of a capacitor to store electric charge.

You can think of a capacitor like a small container that holds electricity. The more it can hold, the greater its capacitance.

It has been found that the amount of charge (we call it Q) stored in a capacitor depends on the voltage (we call it V) across its two plates. When the voltage increases, the charge also increases. This means:

Q is directly proportional to V,
or simply written as:
Q = C × V

Here, C is a constant. We call this constant the capacitance.

Definition of Capacitance

The capacitance of a capacitor is the amount of charge stored on its plates divided by the voltage across the plates.

So,
Capacitance (C) = Charge (Q) ÷ Voltage (V)

Unit of Capacitance

Let’s understand it in a very easy way.

We know that:
Capacitance (C) = Charge (Q) ÷ Voltage (V)

The unit of charge is 1 coulomb, and the unit of voltage is 1 volt.
So, the unit of capacitance becomes 1 coulomb ÷ 1 volt, which is called 1 farad.
The symbol of farad is F, named after the great scientist Michael Faraday.

So we say:
1 farad = 1 coulomb / 1 volt

Now what does this mean?

It means if we apply 1 volt of electric pressure between two plates of a capacitor, and 1 coulomb of charge collects on each plate, then the capacitor has a capacitance of 1 farad.

Let’s take a simple example:
Suppose only 0.1 coulomb of charge is collected when 10 volts are applied.
Then the capacitance will be:

C = 0.1 ÷ 10 = 0.01 farad

But 1 farad is a very big unit, and most capacitors we use in real life are much smaller.
So, we use smaller units like:

• Microfarad (µF):
  1 µF = 1 millionth of a farad = 10⁻⁶ F
• Picofarad (pF):
  1 pF = 1 million-millionth of a farad = 10⁻¹² F

These smaller units are easier to use in circuits and electronic devices.

Factors That Affect Capacitance

Capacitance means how much electric charge a capacitor can store. Some things can change how much a capacitor can hold. Let’s understand them one by one in a very simple way.

Size of the Plates
If the plates of the capacitor are big, it can hold more charge. That means the capacitance will be more.
But if the plates are small, it can’t hold much charge. So, the capacitance will be less.
Big plates = more charge = more capacitance
Small plates = less charge = less capacitance

Distance Between the Plates (Thickness of the Gap)
If the plates are very close to each other, the capacitor can store more charge. So, the capacitance becomes more.
But if the plates are far apart, the capacitor stores less charge, and the capacitance goes down.
Smaller distance = more capacitance
Bigger distance = less capacitance

Type of Material Between the Plates (Dielectric)
Between the plates, there is a special material called a dielectric. If this material is good at holding electric fields, the capacitor can store more charge.
Different materials work differently. Some increase the capacitance more than others.
Better material = more capacitance
Weaker material = less capacitance

In simple words:

• Big plates, small gap, and a good material in between help a capacitor store more charge.
  That’s how we get high capacitance.

Dielectric Constant (or Relative Permittivity)

When we place an insulating material between the two plates of a capacitor, we call that material a dielectric. It does not let electric current pass through, but it helps store electrical energy.

Now, when we charge a capacitor, an electric field is created between the plates. This field passes through the dielectric. The dielectric increases the number of electric lines of force between the plates. As a result, the capacitor can now hold more electric charge.

The strength of this effect — how well a material can increase the number of electric lines of force — is called its dielectric constant or relative permittivity.

To make things simple, we say air has a dielectric constant of 1. This means air is our reference point. If we use another material like mica, which has a dielectric constant of 6, it means mica is 6 times better than air at helping the capacitor store charge.

In other words, if a capacitor can hold a certain amount of charge with air, it can hold 6 times more charge with mica, using the same voltage.

Let’s Understand with a Simple Example

• Suppose:
  • V is the voltage across the plates.
  • With air as dielectric, the capacitor holds Q amount of charge.
  • So, Capacitance with air (C_air) = Q / V
• Now, if we replace air with mica, and keep the same voltage:
  • The capacitor can now hold 6Q charge.
  • So, Capacitance with mica (C_mica) = 6Q / V = 6 × C_air

That means:

Dielectric constant = Capacitance with dielectric / Capacitance with air

So, the dielectric constant tells us how much more charge a material allows a capacitor to store compared to air.

Key Points to Remember

• A dielectric is the insulating material between capacitor plates.
• It increases the charge a capacitor can hold.
• Dielectric constant is a number that tells us how strong the material is at doing this.
• Air has a dielectric constant of 1.
• Other materials like mica have a higher value (e.g., 6), which means more capacitance.

Capacitance of an Isolated Conducting Sphere

Let’s imagine a round metal ball — a perfect sphere — floating alone in space. This ball is not connected to anything else, just sitting in the air or in another material. This kind of setup is called an isolated conducting sphere.

Now, suppose we give this metal ball a positive charge, let’s say +Q. This charge spreads out evenly all over the surface of the ball. That means every part of the outer surface gets the same amount of charge.

To understand how this ball holds the charge, we talk about a word called capacitance.

What is Capacitance?

Capacitance is just a way to tell us how much charge something can store for a certain voltage. Think of it like a bucket. A bigger bucket can hold more water. In the same way, a bigger sphere can hold more charge.

The formula to find the capacitance (C) of this sphere depends on:

• Its radius (r) — how big the ball is,
• And the material around it — called the medium.

Important Formulas

If the sphere is in air:

C=4πε0r

(ε₀ is a constant that tells us about how electric fields behave in air or empty space.)

If the sphere is in a different material (like oil or glass):

C = 4π ε₀ εᵣ r

(εᵣ is the relative permittivity of that material, which shows how much better it is at storing electric charge than air.)

Some Simple Things to Remember:

1. Bigger sphere, more charge:
   The larger the radius, the more charge it can hold. So, a big sphere can store more electricity than a small one at the same voltage.
2. The unit of ε₀ (called “epsilon naught”) is farads per meter (F/m).
   It helps us understand how well space (or air) can allow electric fields to exist.

Example 6.1 –

Question:
27 gol drops, har ek ka radius 3 mm hai aur har drop par 10⁻¹² coulomb charge hai. Jab in sabhi drops ko mila kar ek bada drop banaya jaata hai, to naye drop ka capacitance aur potential kya hoga?

Solution:
Let:

• Chhoti drop ka radius = r = 3 mm = 3 × 10⁻³ m
• Badi drop ka radius = R
• Capacitance = C
• Charge = Q = 27 × 10⁻¹² C (kyunki total charge sabhi drops ka jod hai)

Step 1: Volume ka formula lagao

Badi drop ka volume = 27 chhoti drops ka total volume
(4/3)πR³ = 27 × (4/3)πr³

π aur (4/3) dono taraf se cancel ho jaayenge:
R³ = 27 × r³

R = ³√(27 × r³) = 3r
Toh, R = 3 × 3 = 9 mm = 9 × 10⁻³ m

Step 2: Capacitance ka formula

Capacitance of a spherical conductor:
C = 4πε₀R
(ε₀ = 8.85 × 10⁻¹² F/m)

C = 4 × 3.14 × 8.85 × 10⁻¹² × 9 × 10⁻³
= 1 × 10⁻¹² F
= 1 picofarad (pF)

Step 3: Potential ka formula

V = Q / C
Q = 27 × 10⁻¹² C
C = 1 × 10⁻¹² F

V = (27 × 10⁻¹²) / (1 × 10⁻¹²) = 27 volts

Final Answer:

• Capacitance = 1 pF
• Potential = 27 V

Capacitance of a Spherical Capacitor

Let’s talk about a spherical capacitor. It’s made of two round hollow metal balls, one inside the other. These balls don’t touch each other at all. The inner one is called Sphere A, and the outer one is Sphere B.

We will look at two cases, but for now, let’s talk about the first case:

(i) When the Outer Sphere is Connected to Earth (Earthed)

Imagine this:

• We have two metal spheres, one inside the other.
• The outer sphere B is connected to the earth, which means its voltage becomes zero.
• We give some positive charge (+Q) to the inner sphere A.

As soon as we give a charge to Sphere A:

• It pulls an equal and opposite charge (–Q) to the inner surface of Sphere B.
• A +Q charge also appears on the outer surface of Sphere B, but since Sphere B is connected to the earth, this +Q charge goes away into the ground.
• So now, only the inner surface of B has –Q, and Sphere A has +Q.

Let:

• rA = radius of inner sphere A
• rB = radius of outer sphere B
• εr = relative permittivity of the medium between them
• ε₀ = permittivity of air or vacuum

Now, we find the potential of Sphere A using physics formulas:

VA = (1 / 4πε₀εr) × Q × (1 / rA – 1 / rB)

Since the outer sphere is earthed:

VB = 0

So, the potential difference (P.D.) between A and B is:

VAB = VA – VB = VA

Now, we can find the capacitance (C) of this spherical capacitor using the formula:

C = Q / VAB

Substitute the formula for VA, and we get:

C = 4πε₀εr × (rA × rB) / (rB – rA)

This is the capacitance when the space between the spheres is filled with a material.

If the space is empty or filled with air, then εr = 1, and the formula becomes:

C = 4πε₀ × (rA × rB) / (rB – rA)

(ii) When the inner sphere is connected to Earth

Let’s understand this step by step.

Imagine you have two round balls—one small (inner sphere) and one big (outer sphere). The smaller ball is kept inside the bigger one, but they don’t touch each other. Now, if we connect the inner ball to the Earth, something interesting happens.

This setup acts like two capacitors joined together side by side (this is called “in parallel”).

There are two parts in this setup:

(a) First Capacitor (CBA)
This one is made between:
– The inner surface of the big sphere (B)
– The outer surface of the small sphere (A)

Its capacitance (how much charge it can hold) is given by this formula:

“CBA = 4πε₀ × (rA × rB) / (rB – rA)”

Here:
– rA = radius of the small sphere
– rB = radius of the big sphere
– ε₀ (epsilon naught) is a constant for air

(b) Second Capacitor (CBG)
This one is made between:
– The outer surface of the big sphere (B)
– And the Earth

Its capacitance is the same as if the big sphere is standing alone in air.
Its formula is:

“CBG = 4πε₀ × rB”

So, the total capacitance of the whole system is:

“C = CBA + CBG”

Note: If nothing is mentioned clearly, we usually assume that the outer sphere is connected to Earth.

Example 6.2:

Question:
There is a spherical capacitor. The air gap between its two layers is 2 cm.
Its total capacitance is equal to a single solid sphere with 1.2 m diameter.
We need to find the radii of the inner and outer surfaces of this spherical capacitor.

Given:

• Thickness between coatings = 2 cm
• Radius of single equivalent sphere = 1.2 m ÷ 2 = 0.6 m = 60 cm

So,
“rB – rA = 2 cm”
“rA × rB = 120 cm²” (because rA × rB = R² = 60²)

Now,
Let’s find (rB + rA)²:
“(rB + rA)² = (rB – rA)² + 4 × rA × rB”
= 2² + 4 × 120 = 4 + 480 = 484
So,
“rB + rA = √484 = 22 cm”

Now we solve:

• rB – rA = 2
• rB + rA = 22

Add them:

• 2rB = 24 → rB = 12 cm
• So, rA = 22 – 12 = 10 cm

✅ Answer:

• Inner Radius (rA) = 10 cm
• Outer Radius (rB) = 12 cm

🌟 Example 6.3:

Question:
A capacitor has two thin metal spheres – one inside the other.
Their radii are:

• Inner = 8 cm
• Outer = 10 cm

The outer shell is earthed, and a charge is given to the inner one.
You need to find:
(i) Its capacitance,
(ii) What happens if the outer shell is removed after the inner shell has 200 V potential?

Step 1: Capacitance of spherical capacitor

Given:
rA = 8 cm = 0.08 m
rB = 10 cm = 0.1 m
ε₀ = 8.854 × 10⁻¹² F/m

Formula:
“C = 4π ε₀ × (rA × rB) / (rB – rA)”

Putting the values:
C = 4 × 3.14 × 8.854 × 10⁻¹² × (0.08 × 0.1) / (0.1 – 0.08)
C ≈ 44.44 × 10⁻¹² F

Step 2: Charge when potential is 200 V

Formula:
“Q = C × V”
Q = 44.44 × 10⁻¹² × 200 = 8888 × 10⁻¹² C

Step 3: Capacitance after removing outer shell

Now only the inner sphere remains.

Formula for isolated sphere:
“C′ = 4π ε₀ × rA”
= 4 × 3.14 × 8.854 × 10⁻¹² × 0.08
≈ 8.88 × 10⁻¹² F

Step 4: New potential

Formula:
“V′ = Q / C′”
= 8888 × 10⁻¹² / 8.88 × 10⁻¹²
= 1000 V

✅ Answer:
(i) Capacitance = 44.44 pF
(ii) New Potential = 1000 V after outer shell is removed.

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