A solenoid carries an increasing current, so its magnetic field B is increasing inside. A circular conducting loop is placed outside the solenoid, where B = 0 at every point on the loop. What does Faraday's law predict?
ANo EMF is induced because B = 0 everywhere on the loop
BNo EMF is induced because the loop is outside the solenoid
CAn EMF is induced because the magnetic flux through the loop's interior is changing, even though B = 0 on the loop itself
DAn EMF is induced only if the loop is a conductor; the electric field outside is zero regardless
Faraday's law states EMF = −dΦ_B/dt, where Φ_B is the flux *through the surface bounded by the loop* — not the field at the loop itself. Even though B = 0 at every point on the loop, the flux through the loop's interior (which passes through the solenoid) is changing. This creates a circulating electric field in the space around the solenoid, including on the loop. This is one of the most counterintuitive results in electromagnetism: the source of EMF is the changing flux *through* the loop, not the field *at* the loop. Option A reflects the common misconception of confusing the field at the loop with the flux through it.
Question 2 Multiple Choice
How does the electric field induced by a changing magnetic flux differ fundamentally from the electrostatic field produced by static charges?
AThe induced field is weaker but has the same field-line structure as the Coulomb field
BThe induced field points radially outward from its source; the Coulomb field curls in closed loops
CThe induced field has closed field lines with no source charges; the electrostatic field begins on positive and ends on negative charges
DThere is no fundamental difference — both are solutions to the same equation
The induced electric field is non-conservative: its field lines form closed loops, never beginning or ending on charges. This is the geometric signature of a curl (∇ × E = −∂B/∂t): field lines close on themselves. The electrostatic Coulomb field is conservative — field lines start on positive charges and end on negative charges, and ∮ E · dl = 0 around any closed path. The two kinds of electric field obey different equations and have different topologies. The induced field is fundamentally new — it is created by time-varying B, not by charges.
Question 3 True / False
The induced electric field from a changing magnetic flux can be nonzero even in a region where the magnetic field itself is zero.
TTrue
FFalse
Answer: True
Yes — this is one of the most important and counterintuitive consequences of Faraday's law. The induced E at a point depends not on B at that point but on the *rate of change of magnetic flux* through any surface bounded by a loop around that point. Outside a solenoid, B = 0, but the changing flux through loops that enclose the solenoid generates a circulating E in the surrounding region. This is directly analogous to the Aharonov-Bohm effect in quantum mechanics, where the magnetic vector potential affects particle phases in field-free regions.
Question 4 True / False
In electrostatics, the line integral ∮ E · dl around any closed path is typically nonzero, since the electric field points outward from charges.
TTrue
FFalse
Answer: False
This is backwards. In electrostatics, ∮ E · dl = 0 around any closed loop — the electrostatic field is conservative, meaning it does zero net work on a charge taken around a closed path. This follows from the fact that the electrostatic field is the gradient of a scalar potential: ∮ ∇V · dl = 0 for any closed path. It is the *induced* electric field (from changing B) that can have a nonzero circulation. This is precisely what makes Faraday's law physically deep: it asserts that time-varying B creates a field with nonzero curl, unlike any electrostatic configuration.
Question 5 Short Answer
Why is the induced electric field described as 'non-conservative,' and what does this mean physically?
Think about your answer, then reveal below.
Model answer: A field is conservative if the work it does on a charge around any closed path is zero — equivalently, if the field can be described as the gradient of a potential. The induced electric field is non-conservative because ∮ E · dl = −dΦ_B/dt, which is generally nonzero. This means the field does net work on a charge traversing a closed loop — it can drive a persistent current in a conductor without any battery. Physically, the 'source' of the field is not separated charges (as in electrostatics) but the changing magnetic flux threading through the loop; the energy comes from whatever is changing the magnetic field.
This distinction matters for understanding why transformers and generators work: the induced EMF is not a potential difference between two points (there is no unique potential function) but a true circulation of the electric field around a loop. The concept of voltage loses its usual meaning in the presence of time-varying B, which is why circuit analysis involving inductors requires care.