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Power dissipation

Electrical power measures how quickly energy is transferred. A resistor converts organized electrical energy into internal energy, usually observed as heating. A source can deliver energy to a circuit or absorb energy from it, as a rechargeable battery does while charging. Power is therefore a signed rate, not merely a positive label attached to every component.

The sign becomes meaningful only after voltage polarity and current direction are defined together. Once that convention is fixed, one equation works for resistors, sources, capacitors, inductors, and more complicated devices.

From energy per charge to power

Voltage is energy transferred per unit charge. If a small charge \(dQ\) crosses an element through a voltage difference \(V\), the differential energy transfer is

\[dE=V\,dQ\]

Divide by time and use \(I=dQ/dt\):

\[P=\frac{dE}{dt}=VI\]

The SI unit is the watt:

\[1\ \mathrm W=1\ \mathrm{J/s}=1\ \mathrm{V\,A}\]

Power is an instantaneous rate. Energy transferred during a finite interval is

\[\Delta E=\int_{t_0}^{t_1}P(t)\,dt =\int_{t_0}^{t_1}V(t)I(t)\,dt\]

A kilowatt-hour is a unit of energy, not power:

\[1\ \mathrm{kWh}=3.6\times10^6\ \mathrm J\]

Passive sign convention

Under the passive sign convention, reference current enters the terminal labeled with positive reference voltage. Then

\(P=VI\) has a direct interpretation:

Reversing both the voltage polarity and the current arrow leaves the product unchanged. Reversing only one reference changes the reported sign. Neither operation changes the physical circuit; it changes the coordinate description.

voltage, current, tolerance = 5.0, 0.01, 1e-12
power = voltage * current
role = (
    "absorbing" if power > tolerance else "delivering" if power < -tolerance else "zero"
)
print(role, f"{power:.3f} W")

The tolerance belongs to reporting and numerical comparison. It must not be used to silently discard physically important small powers.

Power in an ideal resistor

For a positive ideal resistance, substitute \(V=IR\) into the general power equation:

\[P=VI=I^2R=\frac{V^2}{R}\]

The squared forms are nonnegative for positive \(R\), so an ordinary passive resistor always absorbs electrical power under the passive sign convention. It does not store that energy indefinitely. Microscopic carrier scattering transfers it into internal energy, and the resistor exchanges thermal energy with its surroundings.

The three expressions describe one operating point, but trend comparisons require a declared constraint. At fixed resistance, doubling current multiplies power by four. At fixed voltage, doubling resistance halves power. At fixed current, doubling resistance doubles power. The last two statements differ because the external source is constrained differently.

Worked DC power ledger

Connect an ideal \(12\ \mathrm V\) across an ideal \(6\ \Omega\). The current magnitude is

\[I=\frac{12\ \mathrm V}{6\ \Omega}=2\ \mathrm A\]

Current enters the resistor's positive terminal, so the resistor absorbs

\[P_R=(12\ \mathrm V)(2\ \mathrm A)=24\ \mathrm W\]

The same loop current leaves the source's positive terminal. If the source branch is still described with passive sign convention, its signed current is negative and

\[P_s=(12\ \mathrm V)(-2\ \mathrm A)=-24\ \mathrm W\]

The circuit power sum is zero:

\[P_R+P_s=24\ \mathrm W-24\ \mathrm W=0\]

This does not mean that no energy moves. It means the rate supplied by the source equals the rate absorbed by the resistor.

Power conservation in a network

For an ideal lumped circuit whose branches use compatible reference conventions, Kirchhoff's laws imply an instantaneous power balance:

\[\sum_k P_k(t)=\sum_k V_k(t)I_k(t)=0\]

Some terms are positive absorptions and others are negative deliveries. The sum provides a strong diagnostic for a solved circuit. A nonzero residual may reveal a reversed polarity, omitted component, mismatched units, stale value, or incomplete convergence.

resistor_power = 5.0 * 0.01
source_power = 5.0 * (-0.01)  # Current enters the negative terminal.
assert abs(resistor_power + source_power) < 1e-12
print("Network power sum (W):", resistor_power + source_power)

A zero sum is necessary but not sufficient. Two equal sign errors can cancel. Also check each element law and Kirchhoff residual separately.

Capacitors and inductors exchange stored energy

An ideal capacitor has

\[I_C=C\frac{dV_C}{dt},\qquad U_C=\frac{1}{2}CV_C^2\]

Its instantaneous power is

\[P_C=V_CI_C =CV_C\frac{dV_C}{dt} =\frac{dU_C}{dt}\]

Positive power increases stored electric-field energy. Negative power returns energy to the rest of the circuit. Similarly, an ideal inductor obeys

\[V_L=L\frac{dI_L}{dt},\qquad P_L=V_LI_L=\frac{d}{dt}\left(\frac{1}{2}LI_L^2\right)\]

Calling all positive branch power “dissipation” would therefore be inaccurate. Resistors dissipate electrical energy into internal energy; ideal reactive elements temporarily store and can return it.

For a transient interval, integrate every branch power using the same timestamps. The change in capacitor and inductor energy should agree with their integrated powers, while the complete source, storage, and dissipative energy ledger should close.

Time-varying and sinusoidal power

For arbitrary time-dependent signals, instantaneous power remains \(p(t)=v(t)i(t)\). Average power over an interval of duration \(T\) is

\[P_{\mathrm{avg}}=\frac{1}{T}\int_{t_0}^{t_0+T}p(t)\,dt\]

For a sinusoidal voltage and current of the same frequency,

\[v(t)=V_0\cos(\omega t),\qquad i(t)=I_0\cos(\omega t-\phi)\]

the average over a whole number of periods is

\[P_{\mathrm{avg}}=V_{\mathrm{rms}}I_{\mathrm{rms}}\cos\phi\]

For sinusoids, \(V_{\mathrm{rms}}=V_0/\sqrt2\) and \(I_{\mathrm{rms}}=I_0/\sqrt2\). A pure resistor has \(\phi=0\) and absorbs positive average power. Ideal capacitors and inductors have zero average power over a complete steady-state cycle even though their instantaneous power alternates in sign.

The compact RMS formula requires sinusoidal steady state and compatible phase information. Multiplying unrelated RMS values is not a replacement for averaging \(v(t)i(t)\) in a non-sinusoidal waveform.

Thermal response and ratings

Electrical dissipation raises temperature only after accounting for heat capacity and heat transfer. A simple lumped thermal model might use

\[C_{\mathrm{th}}\frac{dT}{dt} =P_{\mathrm{diss}}-\frac{T-T_{\mathrm{amb}}}{R_{\mathrm{th}}}\]

This thermal capacitance and thermal resistance are not the electrical capacitance and resistance. The equation says dissipated power warms the component, while heat flow to the environment cools it. Because electrical resistance can itself depend on temperature, electrothermal feedback may occur.

A component's wattage rating is an operating limit under specified cooling conditions, not an extra term in \(P=VI\). Brief pulses, ambient temperature, airflow, mounting, and maximum voltage can all matter. The physical safety question cannot be answered from nominal resistance alone.

Energy integration in Python

For sampled transient data, a trapezoidal update is usually more accurate than treating power as constant at only the start of each interval.

import numpy as np

times = np.linspace(0, 2, 1001)
voltage = 5 * np.sin(2 * np.pi * times)
current = voltage / 100
energy = np.trapezoid(voltage * current, times)
print(f"Heat produced: {energy:.3f} J")

All signals must refer to the same element, orientation, and time basis. Refine the sample interval and compare integrated energy at the same final time. A coarse trace can miss narrow switching losses even when its displayed waveform looks smooth.

Verification and limits

Check that volts times amperes gives watts and watts times seconds gives joules. For a positive resistor, \(I^2R\) must not be negative. Doubling current at fixed resistance must quadruple power. Reversing both branch references must leave power unchanged. A closed ideal network must balance instantaneous power within a scale-aware tolerance.

Ideal accounting omits electromagnetic radiation, unmodeled parasitic elements, chemical and mechanical processes inside sources, measurement loading, and spatial temperature gradients. Numerical energy integrals also inherit sampling and integration error. State the system boundary clearly: energy that seems lost from an electrical subsystem may simply have crossed into a thermal, mechanical, optical, or chemical one.

A worked Python notebook

Drive current of either sign through a 100 ohm resistor. The question for this calculation is precise: how does dissipated power change when we change resistor current, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

Passive resistance absorbs power for either current direction, with the voltage sign following the current. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from -0.1 to 0.1 A. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for resistor current, expressed in A. The output is dissipated power, expressed in W. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 100 q^{2}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 0.05 A, the expression gives 0.25 W. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("100*q**2", locals={"q": q})
reference_input = 0.05
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "A")
print("Predicted dissipated power:", reference_output, "W")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 1 W. The midpoint, at 0 A, gives 0 W. The upper endpoint gives 1 W. Read these together with the assumptions: they describe different choices of resistor current within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([-0.1, 0.0, 0.1])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} A -> {output_value:.6g} W")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows resistor current; the vertical axis shows dissipated power. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(-0.1, 0.1, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Resistor current (A)")
ax.set_ylabel("Dissipated power (W)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does dissipated power change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.05 A is 10 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 200 q\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 0.051000000000000004
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "W")
print("Full expression:", full_prediction, "W")
print("Approximation difference:", full_prediction - linear_prediction, "W")

Here the changed input is 0.051 A. The tangent estimate is 0.26 W, while the full expression gives 0.2601 W. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 0.2
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of resistor current inside the stated interval, predict whether dissipated power will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: Passive resistance absorbs power for either current direction, with the voltage sign following the current. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of W: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(0.1) - evaluate(-0.1))
for points in [101, 201, 401]:
    grid = np.linspace(-0.1, 0.1, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

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