1.1: Voltage

Explore

Examine the Voltage vs. Length Diagrams (right).

1. What is the x-axis label and units? What is the y-axis label and units?
2. Look at the part of the graph with a positive slope. Does the voltage rise or fall during this part of the circuit? What circuit element (look below the graph) causes this rise or fall?
3. Look at the part of the graph with a negative slope. Does the voltage rise or fall during this part of the circuit? What circuit element (look below the graph) causes this rise or fall?
4. Use highlighters or colored pencils to highlight the horizontal parts of the graph using different colors for different voltage values.
5. What parts of the physical circuit correspond to the horizontal segments of the voltage graph? Use the same highlighter colors to color the circuit schematic to show which parts of the circuit correspond to those segments on the graph.

Critical Thinking & Concept Invention

1. Starting at point A, tell a story that identifies what circuit element is between each lettered point and whether the voltage increases, decreases, or does not change as it goes through that circuit element.
2. Identify patterns. How do the different components of a circuit affect the voltage in a circuit? Create an organized summary on your whiteboard to present the key ideas to the class.

Apply

Extend your voltage patterns to the circuit below that has multiple paths.

• Copy the circuit onto your whiteboard.
• Make the wires in the circuit different colors to show different values of voltage.
• Decide as a group if color changes or does not change as you pass through a junction. Make this statement on your whiteboard:

Color (Voltage) does change / does not change through junctions.

Explore

1. Use the information from the model to fill out the table.

Critical Thinking & Concept Invention

1. Identify patterns. As the voltage drops from the battery voltage to zero:
   • Circuit elements with more resistance have greater / lesser voltage drops.
   • A circuit element with no resistance would be:
     1. a flat line
     2. a vertical line
     3. a line slanted down
     4. a line slanted up
2. Part to Whole: Proportionally, what fraction of the total resistance is each resistor? Write an equation to calculate this for each resistor using variables first, then use the resistance values to calculate fractions below each equation.

3. Compare these fractions to the voltage drops for each resistor. How could you use the fractions to calculate the voltage drops if they were not given on the voltage graph?
4. Generalize: Create an equation to determine the voltage drop for different resistors along the same path.

Note: This equation only holds true for resistors in series along a path without a junction.

Apply

A circuit is constructed with a 16 V battery, a 100 \(\Omega\), 160 \(\Omega\), and 220 \(\Omega\) resistor. There are no junctions.

• Draw the circuit. Choose any location for the battery and resistors. Label them with their values.
• Draw a Kirchhoff path that will make a voltage rise for the battery and a voltage drop for the resistors.
• Use the voltage splitting equation to determine the voltage drop for each resistor.
• Draw a Voltage vs. Length diagram (the x-axis will be qualitative).

Model

Explore

1. How many circuit elements (batteries or resistors) exist along the loop \(abcda\) in Circuit A?
2. Underneath the loop equation \(\sum \Delta V_{abcda}=0\), write a \(\Delta V\) for each circuit element on the left side of the equation. Keep the right side equal to zero.
3. Draw two loops on Circuit B and label them based on the lettered points those loops pass through, similar to the loop on Circuit A:
   • Make the first loop go through the battery and resistor \(R_2\).
   • Make the second loop go through the battery and resistor \(R_3\).
4. Label subscripts on the two loop equations that include the lettered points that each separate path goes through. (Some letters might appear in both subscripts.)
5. How many circuit elements (batteries or resistors) exist along each of your loops in Circuit B?
6. Underneath the loop equations, write a \(\Delta V\) for each circuit element on the left side of each equation. Keep the right side equal to zero.

Critical Thinking & Concept Invention

1. Kirchhoff’s Voltage Rule is often stated as:

\(\sum \Delta V_{closed\ loop}=0\)

Explain what this means using examples from the model and your additions to the model.

2. Explain how to generate voltage equations from a circuit using this rule.
3. Some students identify all circuit elements in the entire circuit (without drawing loops) and set the sum of all voltages equal to zero. Does this always work? Explain why or why not using examples from the model and the equations you created.

Apply

Analyze the circuit below by completing the following:

Image Credit: OpenStax University Physics Vol 2 (https://openstax.org/details/books/university-physics-volume-2)

• Draw two Kirchhoff loops and construct a voltage equation for each closed loop.
• Put (+) and (-) signs above each \(\Delta V\) term in your equations.
• For each equation, fill in the battery values on a new line.
• Decide as a group whether the voltage splitting equation can be used here to solve for the voltage drops across each resistor. Explain why or why not.
• If you conclude that the voltage splitting equation can be used, determine the voltage drop across each resistor.

Explore

1. Draw a loop that is counterclockwise on this circuit.
2. Examine each battery:
   • If the path is in the direction from positive to negative in the battery, put a plus sign.
   • If the path is in the direction from negative to positive, put a negative sign next to the battery.

Critical Thinking & Concept Invention

1. Analyze the circuit:
   • Create a Kirchhoff Voltage Rule equation for the circuit.
   • Add the battery voltages into that equation (include the appropriate + and - signs). What is the total voltage drop for all resistors?
   • Use the voltage splitting equation to determine the voltage drop for each resistor.
   • Construct a Voltage vs. Position graph next to the circuit schematic in the model. Show the symbols of the circuit elements underneath the x-axis and label each voltage rise or drop.
2. Summarize: How do batteries affect the voltage in a circuit? Be sure to discuss when and why voltage might increase or decrease as a result of a battery.