Electric Potential: Voltage and Energy

Lesson 4 of 5: Electrostatics

In this lesson:

• Define electric potential energy and voltage
• Calculate and work
• Describe equipotential surfaces and their relation to field lines

What You Will Learn Today

By the end of this lesson, you will:

1. Define electric potential energy and potential
2. Calculate
3. Find work done:
4. Describe equipotentials and their link to field lines
5. Apply conservation of energy to electric problems

What Does "9 Volts" Actually Mean?

• A 9 V battery: energy per unit charge between terminals
• A ball on a shelf has gravitational PE — falls when released
• A charge near another charge has electric PE — moves when released

Today: build tools to measure this stored energy

Electric Potential Energy: Like Gravity

• Lifting a mass against gravity → gravitational PE increases
• Moving against the electric force → electric PE increases
• Release a charge → PE converts to KE

Same conservation framework as mechanics

Gravity Versus Electric Potential Energy

• Both systems: stored energy converts to motion when released
• Direction of "falling" differs — but the math is the same framework

Which Direction Does a Released Charge Accelerate?

• Positive charge released near another positive charge: repulsion → accelerates away
• Positive charge released near a negative charge: attraction → accelerates toward
• Negative charge released near a positive charge: attraction → accelerates toward

In each case: PE decreases and KE increases

Check-In: Which Way Does It Move?

A proton () is released near a fixed proton.

Which direction does it move?

A) Toward — attracted by opposite charges

B) Away — repelled by like charges

C) Still — forces cancel

Think before the next slide…

Answer: Like Charges Repel Outward

Answer: B — Away from the fixed proton

• Both protons are positive → repulsive Coulomb force
• Force pushes the released proton outward
• Moving away → electric PE decreases, KE increases
• Energy is conserved:

From Field to Potential Energy to Voltage

• : force per unit charge at each point in space
• : stored energy of a specific charge at a position
• : energy per unit charge — a property of the field itself

Voltage is independent of which charge you place in the field

Electric Potential : Energy per Unit Charge

• Unit: volt (V) = J/C
• is a scalar — has sign, no direction
• For a point charge at distance :

• → ; →

Potential Near Positive and Negative Charges

• Near : , large close up, decreasing with distance
• Near : , large magnitude close up, shrinking to zero far away

Potential Difference and the Volt

• A voltmeter measures between two points
• A 9 V battery maintains between terminals
• Work moving charge through :

• Positive : agent pushes against the electric force

Worked Example: Work to Move a Charge

Move : from to (source: )

Step 1: Calculate at each point ()

Worked Example: Work Calculation (continued)

,

Step 2: Calculate work

→ agent pushes against repulsion (like compressing a spring)

Check-In: What Is the Sign of Work?

moves from to .

Sign of work done by the agent?

A) Positive — higher potential = positive work

B) Negative — negative charge loses energy at higher

C) Zero — no work needed

Apply

Answer: The Agent Does Negative Work

Answer: B — Negative work

• Negative charge loses energy moving to higher potential
• Electric force pulls toward higher → agent does negative work

From Voltage to the Landscape of Potential

• We know how to calculate at a point
• We know how to use to find work
• What does the full potential landscape look like in space?

Equipotential surfaces give us the answer — and they connect directly to field lines

Equipotential Surfaces: Same Everywhere

• An equipotential surface has constant throughout
• Along an equipotential: , so
• No work done — the electric force is perpendicular to motion
• Around a point charge: equipotentials are concentric spheres

Field Lines Are Perpendicular to Equipotentials

• points in the direction of steepest potential decrease
• is always perpendicular to equipotential surfaces
• Close equipotentials: large over small distance → strong field
• Spread equipotentials: small over large distance → weak field

Field Lines and Equipotentials: Two Configurations

For a point charge: spherical equipotentials. For parallel plates: planar equipotentials.

The Topographic Map Analogy for Potential

• Topographic maps: contour lines connect equal elevation
• Steep terrain → closely spaced contours; gentle slope → spread out
• Equipotentials = electric contour lines; field lines = downhill directions

Rule: contour lines and field lines are always perpendicular

Check-In: Reading the Potential Map

Equipotentials: to are 5 cm apart; to are 1 cm apart.

Where is the field strongest?

A) Between 100 V and 200 V

B) Between 200 V and 300 V

C) Equal in both regions

Which spacing means a steeper slope?

Answer: Closely Spaced Equipotentials Mean Stronger Field

Answer: B

• Same in both regions — but over different distances
• 100 V over 5 cm → gentle slope → weaker field
• 100 V over 1 cm → steep slope → stronger field

Sketching Equipotentials From Field Lines

• Rule: Each equipotential must be perpendicular to every field line it crosses
• High near ; low near ; somewhere between them

Key Takeaways: Electric Potential and Voltage

✓ — scalar; sign of gives sign of

✓ — check signs of both and

✓ Equipotentials ⊥ field lines; closer = stronger field

is scalar; is vector — equipotentials have no arrows

Watch Out: Common Errors with Potential

Higher ≠ more energy — for , higher means lower

along equipotential — the electric force still acts; it's perpendicular to motion

at infinity — never reaches zero at finite from a point charge

Coming Up: Capacitors Store Energy

Lesson 5: Capacitors and Electric Energy Storage

• Two plates at different → energy stored in the electric field
• Energy stored:

Bridge to circuits: here is the same voltage that drives current in Ohm's Law