$$\require{cancel}$$

# 5.3: Steady-State Energy Density Model Applied to Electric Circuits

• • Contributed by Dina Zhabinskaya et al.
• Physics at UC Davis

We will develop conservation of energy relations for electricity that are analogous to those we just developed for flowing fluids. Instead of a real fluid flowing, current electricity (as opposed to static electricity) involves the flow of electric charge. To create intensive energy systems we divide by electric charge, rather than by volume as we did for fluids.

# Summary of the Components of Our Energy-Density Model

We begin by summarizing the components of the steady-state energy-density model we developed in the context of fluids and which we will now generalize to the flow of electric charge. The complete energy-density equation(5.3.1) as applied to fluid phenomena,

$\Delta (\text{total head}) = \frac{E_{pump}}{vol} – I R \tag{5.3.1}$

says that the change in the total fluid energy-density as we move from one point to another point in the stead-state flow will increase due to energy added by a pump and will decrease due to the transfer of fluid energy-density to thermal energy-density. There are four fundamental constructs in this relation: (1) change in the fluid energy-density, (2) fluid flow rate, (3) resistance to fluid flow, and (4) pumps, sources of fluid energy-density. We now look at the counterparts of these four constructs for the flow of electric charge.

## (1) Electric Energy-Density System

In one way, current electricity is simpler than dissipative fluid flow. With fluids we have three energy-density systems that all contribute to the total head. In current electricity, there is only one energy system: the electric potential energy per charge. (Because the mass of charge carriers is typically so small and velocities are small, both the gravitational potential energy changes and the KE changes are totally negligible compared to the changes in the electric potential energy.) The electric potential energy is analogous to the gravitational potential energy we encountered previously. Both the gravitational potential energy and the electric potential energy depend on the amount and separation of something, mass in the gravitational case and charge in the electric case. By dividing by electric charge, we turn the extensive electric potential energy, which depends on the amount of charge, into an intensive quantity. The electric potential energy per charge is given the name electric potential. It is customary to omit the word electric, so frequently we will simply refer to the potential. If we are dealing with electricity, you will know that it is referring to electric potential.

Electric potential has SI units of volts, abbreviated V. Electric charge has units of coulombs, abbreviated $$C$$. Since electric potential must have units of energy per charge, the volt must be a joule per coulomb.

$electric~ potential = \frac{electric~potential~ energy}{charge}$

$volt = \frac{joule}{coulomb}$

$V = \frac{J}{C} \tag{5.3.2}$

Electric potential is commonly referred to as “voltage.” You should get into the habit of always consciously thinking “energy per charge” when you hear or use the term “voltage.”

## (2) Flow of Electric Charge: Current

The unit of current, the quantity of what is flowing past a particular point per second, will be charge per time: coulombs per second. The unit of electric current is the amp, abbreviated A.

$current = \frac{electric ~charge}{time}$

$amp = \frac{coulomb}{second}$

$A = \frac{C}{s} \tag{5.3.3}$

There is an important point we need to get very clear about right from the start. Sometimes the electric charge that flows is associated with a flow of electrons; other times it is associated with the flow of ions, either positive or negatively charged, or both. When we take an energy system approach, we don’t need to know the details. In fact, we don’t want to get bogged down in messy questions, such as: “Just how do electrons move through a material?” Now that is indeed a very interesting question, but it is not the question we are addressing here. As always, with an energy-system approach, we focus on changes in energy, not on the details of the interactions. So, for our purposes now, what flows in current electricity is electric charge. In this kind of analysis, we don’t usually care whether the charge is associated with electrons, protons, ions, or “holes”.

Historically, positive charge was defined in a way that makes the charge on an electron negative. Now, when we speak of current as being in a particular direction, we mean positive charge flow. So, if that charge flow is due to the motion of electrons, then those electrons are in fact moving in the opposite direction. We will always emphasize charge flow, not the flow of the charge carriers, e.g., the electrons, ions, or whatever, when using the steady-state energy density model with electrical phenomena.

## (3) Resistance to Electric Charge Flow

Resistance to the flow of a fluid causes a transfer of energy from the fluid density systems to thermal systems. Likewise, in electric circuits, resistance to the flow of charge causes transfer of electric potential energy to thermal systems. In both cases the amount of energy transferred per unit of transported quantity is equal to the product of the current and the resistance. That is, $$\Delta$$Eth/charge = IR. The unit of electrical resistance is the ohm with abbreviation $$\Omega$$ .

## (4) Sources of Electric Energy per Charge

Batteries and generators in electric circuits are analogous to pumps in fluid systems. Batteries convert chemical energy (bond energy) into electric potential energy. Generators convert mechanical energy (often from water or steam) into electric potential energy through a process involving changing magnetic fields, which we will study in Part 3. Historically, these were called sources of electromotive force abbreviated emf. An upper-case script letter $$“ \varepsilon ”$$ is usually used as the symbol for emf. Electromotive force, $$\varepsilon$$ , is analogous to Epump/vol; it is an energy “per charge.” Thus, the unit for $$\varepsilon$$ is the volt, just like for electric potential. Common practice today is to speak of “voltage” instead of emf when referring to batteries and generators. Thus one commonly hears phrases such as, “The voltage of a ‘D’ battery is 1.5 volts, the same as a ‘double-A’ battery.”

# The Complete Energy-Density Equation for Electric Circuits

Using the four electric components just discussed, the complete energy-density equation(5.3.4) for electric charge becomes

$\Delta V = \varepsilon – I R \tag{5.3.4}$

$where~ ~~~~IR = \frac{\Delta E_{th}}{charge}$

The meaning of this equation(5.3.4) is completely analogous to the meaning of the complete energy density equation(5.3.1) used for fluid flow phenomena. It says that the change in the electric potential energy per charge, or voltage, as we move from one point to another point will increase due to energy added by a battery or generator and will decrease due to the transfer of electric potential energy per charge to thermal energy systems.

The arguments we made in developing the fluid version of the energy-density equation(5.3.1) apply to current electricity as well. If there are no sources or energy transfer into or out of the electric charge system, then the electric potential does not change. But if we attach batteries or generators, we put energy into the system. If there is a current and charge flows through conductors that have resistance, then electric potential energy per charge will be converted to thermal energy, which decreases the electric potential.

As with fluid circuits, we must always remember that the complete energy-density equation(5.3.1) applies to two specific points along the current path. The algebraic sign of the term “IR” also works the same way. If we move in the direction of positive charge flow, i.e., in the direction of the current, then “IR” is positive, and the minus sign insures that the electric potential decreases in energy as we move in that direction. This is often referred to as a “voltage drop” or “IR drop.”

# Power Relationships

The power relationships for current electricity are completely analogous to those for fluids.

$P = \Delta V I \tag{5.3.5}$

rate of change of the electric potential energy system

$P = \varepsilon I$

rate energy is transferred into the electric potential system by a battery or generator

$P = I^2R = \frac{( \Delta V)^2}{R}$

rate energy is transferred into the thermal system from the electric potential system

Example: Calculating Power Dissipation and Current: Hot and Cold Power

(a) Consider the examples given in 20.3 and 20.4. Then find the power dissipated by the car headlight in these examples, both when it is hot and when it is cold.

Strategy

For the hot headlight, we know voltage and current, so we can use $$P = IV$$ to find the power. For the cold headlight, we know the voltage and resistance, so we can use $$P=V^{2}/R$$ to find the power.

Solution

Entering the known values of current and voltage for the hot headlight, we obtain $P = IV = \left(2.50 A\right)\left(12.0 V\right) = 30.0 W.$ The cold resistance was $$0.350 \Omega$$, and so the power it uses when first switched on is $P = \frac{V^{2}}{R} = \frac{\left(12.0 V\right)^{2}}{0.350 \Omega} = 411 W.$

Discussion

The 30 W dissipated by the hot headlight is typical. But the 411 W when cold is surprisingly higher. The initial power quickly decreases as the bulb’s temperature increases and its resistance increases.

(b) What current does it draw when cold?

Solution

The current when the bulb is cold can be found several different ways. We rearrange one of the power equations, $$P = I^{2}R$$, and enter known values, obtaining $I = \sqrt{\frac{P}{R}} = \sqrt{\frac{411 W}{0.350 \Omega}} = 34.3 A.$

Discussion

The cold current is remarkably higher than the steady-state value of 2.50 A, but the current will quickly decline to that value as the bulb’s temperature increases. Most fuses and circuit breakers (used to limit the current in a circuit) are designed to tolerate very high currents briefly as a device comes on. In some cases, such as with electric motors, the current remains high for several seconds, necessitating special “slow blow” fuses.

# The Energy-Density Equation for Both Fluids and Electric Charge

The energy-density equations for fluids and current electricity (without pumps or batteries) are:

$\Delta (total~ head) =~ – I R~~~~~~~~~(\text{fluids}) \tag{5.3.6}$

$\Delta V =~ – IR ~~~~~~~(\text{current electricity}) \tag{5.3.7}$

Up to this point we have emphasized the origin of these equations as residing in the fundamental principle of conservation of energy. However, they are examples of a general class of transport phenomena. Making the current the focus of the equations we have:

$I =~ – \Delta(\text{total head}) \Big(\frac{1}{R}\Big)~~~~~~~~~(\text{fluids}) \tag{5.3.8}$

$I =~ – \Delta V \Big(\frac{1}{R}\Big )~~~~~~~~~~~~~(\text{current electricity}) \tag{5.3.9}$

We interpret these relations to mean that a current of something exists because there is a gradient in the “driving potential” for that something. (Gradient means change in the quantity with change in position.) For fluid flow there must be a gradient in the total head. In order to have electric charge flow, there must be a gradient in the electric potential. In each case, the flow is proportional to the inverse of the resistance.