A long straight wire carries current in the +x direction. At a point directly above the wire (displaced in the +y direction), what is the direction of the magnetic field?
AIn the +y direction, pointing radially away from the wire (like an electric field from a positive charge)
BIn the +x direction, parallel to the current
CIn the +z direction, perpendicular to both the current direction and the radial direction
DIn the −y direction, pointing toward the wire
By the Biot-Savart law, dB ∝ dℓ × r̂. Here dℓ = x̂ (current direction) and r̂ = ŷ (from wire toward field point). The cross product x̂ × ŷ = ẑ, so B points in the +z direction. This perpendicularity — the field wrapping around the wire rather than pointing radially — is the fundamental character of magnetic fields. Unlike electric fields from charges (which point along r̂), magnetic fields from currents always point perpendicular to the radial direction.
Question 2 Multiple Choice
Why does the magnetic field from an infinite straight wire fall off as 1/r rather than 1/r² like the electric field from a point charge?
ABecause magnetic fields are fundamentally weaker than electric fields at any given distance
BBecause the 1/r² in Biot-Savart is less accurate than Coulomb's law for extended sources
CBecause integrating contributions from an infinite number of current elements (each falling off as 1/r² from Biot-Savart) over the full length of the wire yields a net 1/r dependence
DBecause the cross product in Biot-Savart adds an extra factor of r in the denominator
Each infinitesimal current element contributes dB ∝ 1/r² (from the Biot-Savart denominator), exactly like Coulomb's law. But integrating over an infinite wire sums these contributions — elements far along the wire are at large distance but still contribute. The geometry of the integration effectively adds one factor of r in the numerator (from the off-axis angle), resulting in a net 1/r dependence for the complete field. This is the same reason a line charge creates a 1/r electric field, while a point charge creates 1/r².
Question 3 True / False
Magnetic field lines always form closed loops — they never begin or end on any source, in contrast to electric field lines which begin on positive charges and end on negative charges.
TTrue
FFalse
Answer: True
This reflects one of Maxwell's equations: ∇·B = 0 everywhere, which means there are no magnetic monopoles. Magnetic field lines have no sources or sinks — they form closed loops encircling currents. Electric field lines, by contrast, begin on positive charges and end on negative charges (∇·E = ρ/ε₀). This topological difference between E and B is one of the fundamental asymmetries in electromagnetism.
Question 4 True / False
The Biot-Savart law gives the magnetic field contribution from a current element as pointing radially away from the current, analogous to how Coulomb's law gives the electric field pointing radially away from a charge.
TTrue
FFalse
Answer: False
This is the key misconception. Electric fields from charges DO point radially along r̂ (Coulomb's law: dE ∝ r̂/r²). Magnetic fields from current elements point PERPENDICULAR to both the current direction dℓ and the radial direction r̂, encoded by the cross product dℓ × r̂. This perpendicularity is not a minor geometric detail — it is the defining character of the magnetic interaction. Magnetic fields always wrap around currents rather than pointing radially outward from them.
Question 5 Short Answer
What does ∇·B = 0 mean physically, and how does it contrast with the corresponding statement for electric fields?
Think about your answer, then reveal below.
Model answer: ∇·B = 0 means magnetic fields have no sources or sinks — there are no magnetic monopoles. Every magnetic field line that enters a region must also exit it; field lines form closed loops around currents rather than beginning or ending anywhere. For electric fields, ∇·E = ρ/ε₀ (Gauss's law): electric fields diverge from positive charges and converge on negative charges. Electric field lines have sources (positive charges) and sinks (negative charges), while magnetic field lines are sourceless. This is one of the deep structural asymmetries between electricity and magnetism.
The contrast between ∇·E ≠ 0 (in the presence of charges) and ∇·B = 0 (always) reflects the absence of magnetic monopoles in nature. Despite extensive searching, no isolated magnetic 'charge' has ever been found. The Biot-Savart law encodes this: its cross product structure guarantees that the resulting field always curls around the current and never points radially outward from it.