Questions: Induced Electric Field: Non-Conservative Behavior

Question 1

A student tries to define a scalar electric potential V(x,y,z) for the induced electric field around a solenoid with increasing current. Why will this fail?

• A. The field is too weak to measure accurately near the solenoid axis
• B. The induced field circulates in closed loops with non-zero closed-line integrals, so no unique scalar potential can exist
• C. Scalar potentials can only be defined for magnetic fields, not electric fields
• D. The induced field exists only inside the solenoid, not in the surrounding space

Answer: (B)

A scalar potential V can be defined for a field if and only if the field is conservative — meaning the line integral around any closed path is zero. Faraday's law says ∮E⃗·dL⃗ = −dΦ_B/dt ≠ 0 when flux is changing. This non-zero closed-loop integral is what makes the field non-conservative and prevents a unique scalar potential from existing. Integrating the induced E along different paths between two points gives different values — the potential is path-dependent and therefore undefined.

Question 2

What is the physical signature that distinguishes induced electric field lines from electrostatic (Coulomb) field lines?

• A. Induced field lines are straight; electrostatic field lines are curved
• B. Induced field lines originate at positive charges; electrostatic field lines originate at negative charges
• C. Induced field lines form closed loops with no beginning or end; electrostatic field lines begin on positive charges and end on negative charges
• D. Induced field lines can only exist inside conducting materials

Answer: (C)

Electrostatic fields originate on positive charges and terminate on negative charges — they have sources and sinks. The induced electric field has no charges as its source; it is created by a changing magnetic flux. Its field lines therefore close on themselves in loops with no beginning or end. This looping topology is the direct physical manifestation of the non-conservative nature of the field and of ∇×E⃗ = −∂B⃗/∂t ≠ 0.

Question 3

An induced electric field can exist in completely empty space, far from any electric charges.

• T. True
• F. False

Answer: True

This is one of the most profound implications of Faraday's law. The induced electric field is generated by a changing magnetic flux — it requires no charges whatsoever. The field appears in the space surrounding a region where magnetic flux is changing, regardless of whether any charges are present. This showed that electric and magnetic fields could generate each other dynamically, ultimately leading to the possibility of electromagnetic waves in free space.

Question 4

Most electric fields can be fully described using a scalar electric potential V, provided we measure it carefully enough.

• T. True
• F. False

Answer: False

Only conservative (electrostatic) fields can be described with a scalar potential V = −∫E⃗·dL⃗. The induced electric field is non-conservative: its closed-loop line integral equals −dΦ_B/dt ≠ 0, so no path-independent scalar function V exists. Induced fields require a vector potential A⃗, where E⃗ = −∂A⃗/∂t. This is one of Maxwell's equations' key insights: the full electromagnetic field requires both scalar and vector potentials.

Question 5

Why does the non-conservative nature of the induced electric field prevent us from assigning a unique electric potential to each point in space?

Model answer: A scalar potential V is defined as V = −∫E⃗·dL⃗ along a path from a reference point. For this to be a well-defined function of position, the integral must be path-independent — any path between two points must give the same value. This holds if and only if the closed-loop integral is everywhere zero (conservative field). The induced electric field has non-zero closed-loop integrals (∮E⃗·dL⃗ = −dΦ_B/dt), so two paths from the same start to the same end can give different values. 'The potential at point P' becomes undefined — it depends on which path you took.

Concretely: moving a test charge around a closed path surrounding a changing magnetic flux region, the field does net work on the charge. This is impossible if potential energy is well-defined — returning to the same point would require returning to the same potential energy. The net work done on a closed path means no consistent potential energy (or potential) can be assigned to each spatial point, forcing the use of vector potentials instead.