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Guide 21-1b. Circuit Conventions, Definitions, and Relationships (expanded version)

This is an expanded version of a previous guide. The present guide includes additional information about resistivity and power.

 To be or not to be (negative, that is) Conventional current is positive current. We now know--unlike in Ben Franklin's day--that electric current is due to the movement of electrons; hence, electrical current is negative current. Ben Franklin didn't know that and assumed the opposite. His convention stuck and so now we conventionally talk about current as the movement of positive charge even though we know otherwise. The physics still works out as long as we make the direction of positive current opposite that of electron current. Generally, when we just say current without an adjective, we mean conventional (positive) current. Note that inside a battery, the direction of positive current is from the negative to the positive terminal while in the circuit outside the battery, positive current goes from positive to negative. The symbol I represents conventional current. That doesn't mean, however, that you can't get negative values for I.  If you solve a circuit problem and I comes out negative, that may just mean that the conventional current is in the opposite direction as you thought. You'll find that the textbook often uses the symbol V to represent potential difference (sometimes called voltage for short). However, it makes more sense to represent potential difference with a symbol that means change, namely ΔV. This represents the difference of potential between two points in a circuit. If the points are a and b, then another way to write this is ΔV = Vba = Vb - Va. This makes it clear that potential difference can be positive or negative. Note that inside a battery, the potential rises from the negative to the positive terminal while in the circuit outside the battery, potential falls from positive to negative. Electrical resistance is always positive.  The definition of electrical resistance is R = -ΔVr/I. How do you get a positive number out of that?  Well, when positive current passes through a resistor, the change of potential across the resistor is negative. So -ΔVr is positive. The symbol P for power represents a positive number.  For a battery, then, P would represent a gain while for a resistor, P would represent a loss.

Definitions
 Quantity Symbol Defining Formula SI Units Notes Current I I = Q/t C/s or A Conventional current is taken to be positive current. Potential Difference ΔV or Vba ΔV = ΔUel /Q V Potential difference is the change in electrical potential energy per unit charge as charge Q moves from point to point in a circuit. This is referred to as voltage and represented by V in the textbook under the assumption that the lower potential is always 0. In your problem solutions and lab work, avoid use of the naked V symbol. Resistance R R = -ΔVr /I Ω Since ΔVr across a resistor is negative (from + to -) and current is positive (from + to -), the quantity R = -ΔVr /I is always positive. Resistivity ρ R = ρL/A Ωm The resistivity characterizes the resistance of a wire of length L and cross-sectional area A. The resistivity depends on the material and may also be influenced by temperature. Temperature coefficient of resistivity α ρ = ρo[1 + α(T - To)] 1/°C The temperature coefficient of resistivity characterizes how the resistivity of a material depends on temperature. In the formula, ρo is the resistivity at some reference temperature, and ρ is the resistivity at a temperature ΔT different from the reference temperature. Power P P = |ΔUel |/Δt W The absolute value of the change of electrical potential energy per unit time is the power. Letting P always represent a positive quantity is conventional. In a battery where electrical potential energy is gained from the negative to positive terminal, P would represent a power gain. In a resistor where electrical potential energy is lost from the positive to negative side, P would represent a power loss. This is sometimes referred to as power dissipation.

Relationships
 Name Formula Notes Conservation of charge The fact that charge is conserved leads to the conclusion that the current is the same in all parts of a single loop circuit. From the definition of current, I = Q/Δt.  For a given Δt, equal charge means equal current. Conservation of energy The total change in electric potential energy ΔUel around a circuit  is 0. All the energy produced by the battery is used in the circuit.  Now ΔUel = QΔV, where ΔV is the difference potential around the circuit. But the difference of potential in returning to the same point must be 0. Another way of saying this is that the algebraic sum of all the potential differences around a circuit must be 0. This is simply an expression of conservation of energy. Ohm's Law ΔVr is proportional to I or ΔVr = -RI, where R is constant Note the difference in the way we express this law from that in the text. Again, we emphasize that there is a difference of potential. We also emphasize the proportionality between the difference of potential across a resistor and the current in the resistor. The constant of proportionality is -R. The negative sign is needed, because ΔVr is negative for positive current in a resistor. Note that Ohm's Law isn't obeyed for many circuit components and hence, doesn't have the stature of, say, the Law of Gravitation or Newton's Laws. Power production in a battery Pb = I |ΔVb| The power gain in a battery is the increase of potential across the terminals multiplied by the current. Power dissipated in a resistor Pb = |ΔVr|2/R = I2R The power dissipated in a resistor can be expressed either as the square of the potential difference across the resistor divided by the resistance or as the square of the current in the resistor multiplied by the resistance. A derivation of this is given below.

More about power in a simple circuit:

Consider the circuit of a battery and resistor shown above.  Positive charge Q passing from D to A in the battery gains electrical potential energy , where is the difference of potential across the battery. The power gain Pb is:

Positive charge Q passing from B to C in the resistor loses electrical potential energy , where is the difference of potential across the resistor. The power loss Pr is:

Here is an alternative derivation for Pr: