Electric Potential Energy: The Backbone of Electrical Systems

Introduction

Hey everyone! Today, let’s dive into a mind-blowing topic that’s all around us but often overlooked – electric potential energy. You might think it sounds complex, but trust me, it’s like the force that keeps the entire world of electrical systems running. Imagine it as the fuel that powers everything from your smartphone to the lights in your home.

What is meant by electric potential energy?

Okay, so first things first – what exactly is electric potential energy? Electric potential energy refers the energy that is needed to move a charge against an electric field. This attraction or repulsion between charges is related to their positions relative to one another. Changes in potential energy can lead to electrical work and various electrical phenomena.

where U is the electric potential energy (in J = joule). Energy is a scalar quantity, so U has no direction. (we use p = |q|r rather than p = qd).

What is the derivation of electric potential energy equation?

Rewriting General Physics equation, the work W_a→b done by a force F in moving a particle from point a to point b is defined to be

If the force is conservative, W_a→b = U_a – U_b, where U is the potential energy associated with that force.
If we have a test charge q₀ outside a spherically-symmetric, same-sign charge q,

,where

and dl cos φ = dr. Then integrating from r_a to r_b gives us

Thus the simplest choice for the potential energy expression is

This equation also holds for opposite-sign charges. Any equation in this block containing ε₀ is for vacuum ≈ air. is the electric potential energy (in J). Recall that 1/4πε₀= 8.988 × 109 N·m² /C² ≈ 9.0 × 109N·m² /C²

q and q₀ are the charges (in C)—(point or spherically symmetric) (recall that charges can be +, 0, or –). r is the center-to-center distance (in m) between the two charges (recall that r is always +). Thus we see that U is + for same-sign charges and – for opposite-sign charges. We also see that U approaches zero as the distance r approaches infinity. This U can be used in K_a + U_a = K_b + U_b (W_other = 0),

The (electric) potential V at a point is the electric potential energy per charge at that point, V =
U/q₀. Therefore, dividing both sides of W_a→b = U_a – U_b by q₀ tells us that the work per charge
W_a→b/q₀ is the (electric) potential difference V_a – V_b between points a and b :

We often shorten V_a – V_b to V_ab (and for electric circuits, often further to just V). V is the (electric) potential (in V = volt = joule/ coulomb = J /C ) (V can be +, 0, or –). W_a→b is the work done (in J) by the external electric field in moving q₀ from a to b (W_a→b can be +, 0, or –).

For a point charge q (or outside a spherically symmetric charge q) we can substitute

to find

For a collection of these charges,

q_i is the charge (in C) of object i (qi is +, 0, or –). r_i is the distance (in m) from the center of object i (r_i is always +). For a continuous charge distribution,

That is, first find an expression for the dV from an arbitrary dq in the distribution, then integrate to find V Finding V_a – V_b from E : Recall that

Canceling out q₀, we have an equation we can use to find the potential difference if we know E all along any path from a to b.

we’ll replace dl with dx or dy or dz, or dr in the integral. Note that E can have units of either V/m or N/C : 1 V/m = 1 (J/C)/m = 1(N·m/C)/m = 1N/C .

What is an example of electric potential energy in real life?

Some of the most common examples of electrical potential energy is when you rub a balloon on your hair and it sticks to a wall, that’s a bit of electric potential energy in action. Weird, right? But cool! The same thing happens when you shuffle your feet on a carpet and get a little zap when touching a metal doorknob.

Basically, it’s all about how charges build up. Like, imagine positive and negative charges as tiny teams of cheerleaders. When they get together on one side of the balloon or your body, they’re like, “Woohoo, we’ve got potential energy!” And when they finally get the chance to jump from the balloon or you to the wall or doorknob, they release that energy in the form of a little electric shock.

Voltage: The Driving Force

Now, let’s talk about voltage. Think of it as the pressure pushing those charges around. It’s like water pressure in a hose – the higher the pressure, the stronger the flow. So, when we talk about high or low voltage, we’re talking about how eager those charges are to move and do their thing.

In the world of electrical systems, voltage is a big deal. It’s what makes electricity move from place to place, just like water flows through a pipe. Without voltage, all those electrical devices we love would be just expensive paperweights!

How is voltage related to potential energy?

Voltage is directly related to potential energy in electrical systems. Voltage represents the amount of energy carried by electric charges per unit of charge, measured in joules per coulomb ( = volts). The Higher the voltage means the greater the potential energy. The greater the potential energy driving charges to move and more “work” can be done, while lower voltage indicates less energy available for electrical devices and circuits.

How Electric Potential Energy Moves

Alright, now you might wonder how this electric potential energy actually moves. The secret lies in conductors and insulators. Conductors are like the superstar carriers of electric charge; they allow electricity to flow through them with ease. Copper wires are a classic example of conductors.

On the other hand, insulators are like the bodyguards of charge; they keep it contained and don’t let it escape. Rubber is a great insulator, which is why electrical cords have rubber coatings.

That’s where closed circuit come into play. A closed electrical circuit is like a never-ending “Ring Around the Rosie” game. The charges can keep going round and round, allowing electricity to flow continuously, powering our gadgets and appliances.

What is work in an electric field?

In an electric field, work refers to the energy transfer that occurs when a charged particle moves between two points with different electric potentials. If the particle moves in the direction of the electric field, work is done by the electric field, increasing the particle’s kinetic energy. Conversely, moving against the electric field requires work to be done on the particle, reducing its kinetic energy.

Suppose that an electric field moves an electron (charge = –e = –1.602 × 10^–19 C) through a potential rise of one volt (V_b larger than V_a by 1 V). Then W_a→b = q0(V_a – V_b) = –e(–1 V) = 1 eV = (1.602 × 10^–19 C)(1 JC) = 1.602 × 10–19 J. That is, one electron volt (1 eV) is defined to equal 1.602 × 10–19 J of work or energy. The term equipotential means constant potential: V is constant at all points along an equipotential line, over an equipotential surface, or throughout an equipotential volume . For equipotential lines or surfaces, Va – Vb = V – V = 0 = ⌡ E cos φ dl tells us that φ must be 90 ̊ (if E ≠ 0). Thus, electric field

lines are always perpendicular to equipotential lines or equipotential surfaces. If its free charges are at
rest overall, a conducting surface is always an equipotential surface with any electric field at its surface normal to that surface. Since the derivative of a constant is zero, E = – ∇ V tells us throughout an equipotential volume, E = 0. The electric field from a charged conducting surface tends to be greatest where the radius of curvature is smallest (for sharp points and thin wires). This large electric field can break down the air, giving an electrical discharge. In contrast, flatter areas tend to have smaller surface electric fields. Finding E from V:

tells us that –dV = E*dl. Evaluating the scalar (dot) product in terms of components gives –dV = E_xdx + E_ydy + E_zdz. Keeping both y and z constant gives dy = 0 and dz = 0, so –(dV/dx )y,z constant = E_x. Such a derivative is called a partial derivative and we have:

E_x is the x-component of the electric field E (in V/m or N/C) (with the same idea for y, z, or r) . Then E = E_x i+ E_y j+ E_z k= –( i ∂/∂x + j ∂/∂y + k∂/∂z )V. The quantity in the parentheses is the gradient operator ∇ , so E = – ∇ V. In words, the electric field equals the negative of the potential gradient

Practical Applications of Electric Potential Energy

Alright, let’s talk about the juicy stuff – how electric potential energy affects our lives. Remember when we talked about how it powers your smartphone? Oh yeah, that little device is like a tiny powerhouse, thanks to electric potential energy. Every time you charge it up, those charges are getting all pumped up with energy, ready to keep your phone running throughout the day.

And what about the lights in your house? It’s not magic that they turn on with a flick of a switch; it’s electric potential energy doing its thing. When you flip that switch, you complete the circuit, allowing the electricity to flow from the power source to the lightbulb.

But it’s not just about gadgets and lights. Think about those massive industrial machines that create everything from cars to the packaging of your favorite snacks. Yep, electric potential energy is at work there too, making sure everything runs smoothly and efficiently.

Safety and Precautions

Now, before we get too carried away with all the wonders of electric potential energy, we need to talk about safety. Electricity is super cool, but it can also be dangerous if we don’t handle it properly.

So, when you see those warning signs or hear grown-ups telling you not to stick your fingers into electrical outlets, they’re not just being buzzkills. They’re looking out for you because touching those outlets when you’re not a trained electrician could give you a shocking experience – and not the good kind!

Conclusion

Alright, my awesome friends, we’ve journeyed through the electrifying world of electric potential energy, and I hope you’re as amazed as I am! From understanding how charges build up and create potential energy to seeing how voltage pushes those charges to move and do their thing, it’s like a never-ending rollercoaster ride of fascination.

Electric potential energy is like the backbone of all electrical systems. It’s the reason we have electricity powering our lives, making everything from our gadgets to our homes come alive. So, the next time you turn on the lights or charge your phone, take a moment to appreciate the marvels of electric potential energy at work.

Just remember to stay safe and be responsible around electricity. It’s a powerful force that we should respect and handle with care. And who knows, maybe one day, you’ll be the one making groundbreaking discoveries in the world of electrical systems! Until then, keep on being curious and exploring the wonders of science all around us. Stay electrified, my friends!

Class Notes