Cylindrical Capacitor: types, formula, and useful exam 4Facts

Cylindrical Capacitor

A cylindrical capacitor is made up of two round, hollow tubes that are placed one inside the other. These tubes are called cylinders. The space between them is filled with a special material that does not let electricity pass through—this is called insulation.

One of these cylinders is in the middle—this is the inner cylinder, and the other one surrounds it from outside—this is the outer cylinder.

This type of capacitor is not just something we see in theory—it is actually used in real life. For example, a single-core cable works just like a cylindrical capacitor. In this cable:

• The core or wire inside acts like the inner cylinder.
• The outer covering or metal layer acts like the outer cylinder.
• And the insulating material in between keeps the electricity from leaking out.

The outer layer is usually connected to the ground (or earth), which helps keep everything safe.

So, in simple words, a cylindrical capacitor is just two round tubes, one inside the other, with insulation in between, and it helps to store electric energy safely.

Capacitance of a Single-Core Cable

Imagine you have a cable that has just one wire in the center. Around this wire, there is an insulating layer, and around that, there is a protective outer cover called the sheath.

• Let’s say the diameter of the wire (the conductor) is d meters.
• The diameter of the sheath (the outer layer) is D meters.
• The material between them is an insulator, and we’ll call its relative permittivity as εᵣ.
• The total electric charge in one meter length of this cable is Q coulombs.

Now, imagine we draw an invisible cylinder of radius x meters between the conductor and the sheath. According to Gauss’s Law, the total electric flux going out of this cylinder is Q.

The surface area of this invisible cylinder is:

“2πx m²”

So, the electric flux density at that distance x is:

“Dx = Q / (2πx) C/m²”

The electric field strength (or electric intensity) at the same point is:

“Ex = Q / (2πε₀εᵣx) V/m”

Now, let’s say we want to move a tiny positive charge from the wire (inner conductor) out to the sheath. The work done in doing that gives us the voltage (V) between them.

So the voltage between the conductor and the sheath is:

“V = Q / (2πε₀εᵣ) × log(D/d)”

This is the potential difference between the inner and outer layers.

Now, the capacitance per meter length of the cable, which tells us how much electric charge the cable can hold per volt, is:

“C = Q / V = 2πε₀εᵣ / log(D/d) F/m”

We can write this in a more specific form using values of constants:

“C = 2.8854 × 10⁻¹¹ × εᵣ / log₁₀(D/d) F/m”

If the cable is l meters long, then the total capacitance becomes:

“C = (2.8854 × 10⁻¹¹ × εᵣ × l) / log₁₀(D/d) F”

Or in a simpler numerical form:

“C = 414 × εᵣ × l / log₁₀(D/d) pF”

Derivation of Cylindrical Capacitor Formula

A cylindrical capacitor has two hollow cylinders – one placed inside the other.
Let the radius of the inner cylinder be ‘a’ and the radius of the outer cylinder be ‘b’, where b > a.
Let the length of both cylinders be L.
We give the inner cylinder a charge of +Q and the outer cylinder a charge of –Q.

Step 1: Use Gauss’ Law to find the electric field between the cylinders

We imagine a cylinder (called a Gaussian surface) placed between the two cylinders.
Let the radius of this surface be r, where a < r < b, and the length of this surface be L.

According to Gauss’ Law,
Electric flux (Φ) = Total charge inside / ε₀

Area of the cylindrical surface = 2πrL

So,
Φ = E × 2πrL = Q / ε₀
⟹ E = Q / (2π ε₀ r L)

Step 2: Use the electric field to find the potential difference (V)

Now,
V = −∫ E dr
⟹ V = −∫[a to b] (Q / 2π ε₀ r L) dr
⟹ V = −(Q / 2π ε₀ L) ∫[a to b] (1/r) dr
⟹ V = −(Q / 2π ε₀ L) [ln(b) − ln(a)]
⟹ V = −(Q / 2π ε₀ L) ln(b/a)

We remove the minus sign because we are taking potential from inner to outer (positive to negative), so:
V = (Q / 2π ε₀ L) ln(b/a)

Step 3: Use the formula of capacitance

We know,
C = Q / V

Now putting the value of V:
C = Q / [(Q / 2π ε₀ L) ln(b/a)]

Q cancels out:
C = 2π ε₀ L / ln(b/a)

Final Formula:
C = (2π ε₀ L) / ln(b/a)

This is the formula for the capacitance of a cylindrical capacitor. It shows how the capacitance depends on the length of the cylinders (L), the distance between them (a and b), and the constant ε₀.

Potential Gradient in a Cylindrical Capacitor (Cable)

When we send electricity through a cable, the insulation around the cable has to handle an invisible force. This force is called dielectric stress. It’s like pressure inside the insulation caused by the electric current.

Now, this dielectric stress is the same as the potential gradient — which simply means how fast the voltage is changing from the center of the cable to its outer covering. Think of it like how steep a slide is: the steeper it is, the faster you’ll go down. Similarly, the sharper the change in voltage, the more stress there is at that point.

📌 Let’s Understand with a Cable Example

Imagine a cable with just one wire inside.

• The wire at the center is called the core (it carries the current).
• Around this core, there is insulation.
• Outside the insulation, there is a sheath (a protective cover).

Let’s say:

• The diameter of the core is d
• The inner diameter of the sheath is D

We want to know: how does the electric pressure (potential gradient) change as we move from the wire (center) out towards the sheath?

📉 How the Potential Gradient Changes

The potential gradient is strongest near the center (at the surface of the wire) and gets weaker as we go outwards toward the sheath.

So:

• At the surface of the core → maximum stress
• At the inner surface of the sheath → minimum stress

This happens because the voltage is spread over a distance — and the farther you go from the center, the less steep the voltage drop becomes.

🔢 Important Points to Remember

1. Potential Gradient is just another name for electric intensity — it tells us how strongly the voltage is changing.
2. The closer you are to the core, the higher this intensity or gradient is.
3. At the core surface (when x = d/2), we get maximum potential gradient.
4. At the sheath surface (when x = D/2), we get minimum potential gradient.

⚠️ Why is This Important?

When designing a cable, we must always make sure that the insulation can handle the highest stress near the core.

If the maximum stress is, say, 5 kV/mm, then the insulation must be strong enough to handle at least 5 kV/mm. If not, the insulation might break down, and the cable can get damaged or even cause a dangerous situation.

Most Economical Conductor Size in a Cable

When we design an electric cable, one of the most important parts is the conductor — the part that carries the electric current. But we have to be careful: too much stress (or pressure) on the cable’s insulation can cause it to fail. This stress is strongest at the surface of the conductor.

To keep the cable working safely, the strength of the insulation (called dielectric strength) must be higher than the maximum stress. If it’s not, the cable could break down.

There is a formula that helps us understand this stress:

Maximum stress,
g_max = 2V / (d × loge(D/d))
(where V is the voltage, D is the inner diameter of the sheath, and d is the diameter of the conductor)

Now, in most cable designs, we cannot change the voltage (V) or the outer sheath diameter (D) — these are fixed based on safety and performance. That leaves us with only one part to adjust: the conductor size (d).

To find the best (or most economical) size of the conductor, we want the stress (g_max) to be as low as possible. After working through the formula, we find that the stress is lowest when:

D/d = e, where e ≈ 2.718

This means the best (most economical) diameter of the conductor is:
d = D / 2.718

When this size is used, the maximum stress is:
g_max = 2V / (2.718 × d)
This stress will always be at the surface of the conductor.

Real-Life Cable Design

In real life, this formula gives a great starting point, but we have to think about more than just stress.

• For low and medium voltage cables, this formula might give us a conductor that’s too small. If the conductor is too thin, it may get too hot when current flows through it. So, for these cables, we choose the size based on how much current it needs to carry safely — not just stress.
• For high voltage cables, this method may give us a very large conductor size. That’s okay, because in these cases, using a bigger conductor actually helps reduce stress and makes the cable safer.

How to Get a Bigger Conductor Without Using Too Much Copper

Using a large copper conductor can be very expensive. But there are clever ways to get the same size without spending too much:

1. Use aluminium instead of copper
   Aluminium is lighter and cheaper. For the same current, an aluminium conductor needs to be a bit thicker than copper, which helps.
2. Use copper wires around a central core
   Instead of a solid copper rod, we can use smaller copper wires wrapped around a core made of materials like hemp (a plant fiber). It gives us the size we need with less copper.
3. Use a lead tube in the center
   Sometimes, a small lead tube is placed in the center instead of hemp. This adds strength and helps increase the conductor size without adding much cost.

Capacitance Between Parallel Wires

Imagine you have two long wires hanging in the air. These wires are straight, side by side, and go on and on—just like the wires you see on electric poles or transmission towers.

Let’s call them Wire A and Wire B.
They are placed at a certain distance apart, and each wire has a round shape (like a pipe) with some thickness.

Now think of this:
One wire has positive charge and the other has negative charge. This means electricity is ready to flow from one to the other, and there is an invisible pull between them.

But something interesting happens here:
Even though these wires are not touching, they affect each other through the air. This invisible effect is what we call capacitance.

💡 What Is Capacitance?

Capacitance simply means how much electric charge the wires can hold or store, based on:

• How far apart they are,
• How thick the wires are,
• And the material between them (in this case, just air).

It’s like this:
When you stretch a rubber band between two fingers, the more you stretch it, the more energy it stores.
Similarly, when charges are stored on two wires, that “stretch” or “pull” between charges becomes energy.
And how much they can store—that’s called capacitance.

📏 What Does It Depend On?

Capacitance depends on:

• Distance (d) between the wires – the more the distance, the less the capacitance.
• Radius (r) of the wire – the thicker the wire, the more it can store.
• Air (or any material) between them – air works fine, but some materials can increase capacitance.

Insulation Resistance of a Cable Capacitor

When we use a cable to carry electricity, it has a metal wire inside called a conductor. Around this conductor, we add a layer called insulation.

Why do we need insulation?
Because it stops the electric current from leaking out. This keeps the cable safe and working properly.

The current always tries to take a shortcut—just like water might leak through a hole. But the insulation blocks the current from escaping. The strength with which the insulation resists this leakage is called the insulation resistance.

Why is insulation resistance important?

If the insulation resistance is very high, the cable is safe and strong.
If the insulation resistance is low, the current might leak, which can be dangerous.

Let’s understand with a simple example:

Imagine a single-core cable. That means it has one conductor inside.
Let:

• The radius of the conductor be r₁
• The inner radius of the outer covering (called the sheath) be r₂
• The length of the cable be l
• The material of insulation has a property called resistivity (ρ) – this tells us how much it can resist current.

We now look at a tiny ring inside the insulation, at a distance x from the center.
If this small ring has thickness dx, the resistance it offers to leakage can be written in a small formula.

When we combine all such small rings across the whole insulation, we get the total insulation resistance of the cable.

The final formula comes out as:

R = (ρ / 2πl) × log (r₂ / r₁)

What does this mean?

• If the cable is longer (l increases), the insulation resistance becomes less.
• If the cable is shorter, the insulation resistance becomes more.

So, long cables have less insulation resistance.
That’s why we need to check the insulation of long cables carefully.

To remember:

✅ Good insulation = High resistance = Safe cable
❌ Poor insulation = Low resistance = Risk of leakage

Where Cylindrical Capacitors Are Used:

1. Power Electronics:
   • For energy storage, filtering, and smoothing in power supplies and converters.
   • Used in inverters, UPS systems, and motor drives.
2. Motors and HVAC Systems:
   • In single-phase motors to provide the phase shift for starting and running (start and run capacitors).
   • Widely used in air conditioners, refrigerators, and fans.
3. Lighting Systems:
   • In fluorescent lamps and LED drivers for power factor correction and filtering.
4. Audio Equipment:
   • Used in audio crossovers and amplifiers for signal filtering and energy storage.
5. Industrial Equipment:
   • For power factor correction in large machines and plants.
   • Used in capacitor banks.
6. Electronics:
   • In radios, TVs, and computers for timing, filtering, and decoupling.
7. Medical Devices:
   • For defibrillators and imaging equipment that need quick energy discharge.

✅ Why Cylindrical Shape?

• Efficient heat dissipation
• Compact and space-saving
• Easy to mount vertically or horizontally
• Uniform electric field inside

Applications of Cylindrical Capacitors

Cylindrical capacitors may look simple, but they play a very important role in many of the things we use every day. Here are some places where they are used:

• Electric cars and solar systems – They help control the flow of electricity, so the power stays steady and safe.
• Speakers and sound machines – They make sure the sound is clean and clear, without any noise.
• Medical machines – Devices like pacemakers and shock machines use them to give quick and safe electric pulses.
• Big factory machines – They help run heavy tools like welders and drills without sudden power drops.
• Electronic circuits – They help block the signals we don’t want, so only the right signals go through.
• Storing energy – They hold a bit of electric charge and give it when needed.
• Keeping voltage smooth – They stop the power from jumping up and down too much, so devices work better.
• Giving steady power – They help machines get the right amount of power at the right time.
• Helping devices live longer – Because of their support, machines run smoothly and don’t get damaged easily.

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