Questions: Potential Difference and Voltage

Question 1

A positive test charge is moved from point B to point A along three different paths through an electrostatic field. Which result should you expect?

• A. The three paths give different work values — longer paths involve more field interaction and therefore more work
• B. All three paths give the same work done by the field, because the electrostatic field is conservative and work depends only on the endpoints
• C. The straight-line path gives the minimum work; curved or longer paths always give more
• D. The work values differ unless the field happens to be uniform

Answer: (B)

Electrostatic fields are conservative — they derive from a scalar potential. A defining property of conservative fields is path independence: the work done on a charge moving between two fixed endpoints is the same regardless of the route. This is not a special case; it holds for any electrostatic configuration. Path independence is what allows potential difference V_A − V_B to be defined as a property of the two points alone, not of the path between them.

Question 2

A student measures 5 V between points A and B with a voltmeter and concludes: 'Point A must be at 5 V.' What is wrong with this reasoning?

• A. Nothing — a voltmeter directly reads the absolute electric potential at each probe
• B. The voltmeter measures only the potential difference between A and B; A could be at 105 V with B at 100 V, or A at 5 V with B at 0 V — any pair differing by 5 V is consistent with the reading
• C. The reading should be divided by the charge of the test particle to get the voltage
• D. Voltmeters measure current; a different instrument is needed for voltage

Answer: (B)

A voltmeter measures potential difference, not absolute potential. Absolute potential at a single point is only defined relative to an arbitrary reference (infinity, ground, or a chosen node). The difference between A and B is always 5 V regardless of that reference choice — that's what the voltmeter captures. This is why circuits use 'voltage across' a component (a potential difference) rather than 'voltage at' a single node (which requires specifying a reference).

Question 3

The work done by the electric field in moving a charge between two points in an electrostatic field depends on the path taken between those points.

• T. True
• F. False

Answer: False

Electrostatic fields are conservative — the work done on a charge moving between two endpoints is path-independent. Whether you take the direct route or a complicated detour, the field does the same work. This follows from the fact that electrostatic fields satisfy ∇ × E = 0, the mathematical statement that they are conservative. Path independence is what justifies defining potential difference as a property of the two endpoints alone.

Question 4

Potential difference between two points is path-independent, meaning any route between the same two endpoints gives the same voltage value.

• T. True
• F. False

Answer: True

This is a direct consequence of the conservative nature of electrostatic fields. The integral V_AB = −∫(B to A) E · dl gives the same result for any path from B to A. This path-independence is what makes potential difference physically meaningful: you can attach a voltmeter between two points in a circuit and get a definite reading, without needing to specify which 'path' the measurement traced through the field.

Question 5

Why is potential difference (voltage) path-independent? What property of the electric field is responsible for this, and why does it matter?

Model answer: Potential difference is path-independent because the electrostatic field is conservative — it is derivable from a scalar potential, and equivalently its curl is zero (∇ × E = 0). For any conservative field, the work done between two endpoints is the same regardless of path, because any closed path returns zero net work. This matters because it allows voltage to be defined as a property of two points alone, not of the route between them. Without path-independence, there would be no well-defined 'voltage across' a resistor or capacitor, and circuit analysis would be impossible.

The conservative property traces back to the Coulomb origin of electrostatic fields: Coulomb's law is a central force law, and central force fields are always conservative. This is why we can define electric potential V at each point in space and get potential difference simply by subtracting: V_A − V_B. The entire framework of circuit analysis — Kirchhoff's voltage law, Ohm's law, capacitance — depends on this path-independence holding.