Rucete ✏ AP Physics C In a Nutshell
14. Magnetic Fields
This chapter introduces the definition of magnetic fields, the forces they exert on moving charges and currents, and the calculation of magnetic fields using Biot-Savart and Ampere’s laws.
What Is a Magnetic Field?
• A magnetic field B is a vector field created by moving charges or intrinsic electron spin.
• Magnetic fields determine the force experienced by moving charges.
• Units: Tesla (T), where 1 T = 1 N/(A·m).
Sources of Magnetic Fields
• Moving point charges and currents in wires generate magnetic fields.
• Permanent magnets arise from microscopic moving charges (electron spin).
Force on a Moving Charge in a Magnetic Field
• Magnetic force: F = qv × B.
• Only moving charges experience magnetic force; stationary charges feel no magnetic force.
• The force is zero if the charge moves parallel or antiparallel to B.
• The magnetic force is always perpendicular to both v and B, affecting direction but not speed.
• Use the right-hand rule: fingers sweep v to B, thumb points force direction for positive charges (reverse for negative).
Net Force with Both Electric and Magnetic Fields
• Total force: F = q(E + v × B).
• Electric and magnetic forces superpose independently.
Magnetic Field Lines
• Field lines are tangent to B, with density proportional to field strength.
• Field lines form closed loops; they do not begin or end (no magnetic monopoles).
• Outside magnets, lines run from north to south poles.
Motion of Charged Particles in Constant Magnetic Fields
• Parallel to B: Moves straight without deflection.
• Perpendicular to B: Undergoes uniform circular motion; centripetal force provided by magnetic force.
• Circular motion radius: r = mv/(qB).
• Combination of parallel and perpendicular components: Motion is a helix with axis along B.
Force on a Current-Carrying Wire
• A wire carrying current I feels force F = I l × B.
• Force direction follows the right-hand rule (current direction is like positive charges).
• In curved wires, divide into small segments and integrate: dF = I dl × B.
Forces on Closed Wire Loops
• In a uniform magnetic field, the net force on a closed wire loop is zero, but torque may be nonzero.
Mass Spectrometer Applications
• In region 1, charged particles gain kinetic energy by passing through a voltage V: KE = qV.
• In region 2, perpendicular electric (E) and magnetic (B) fields act; only particles with velocity v = E/B pass undeflected.
• In region 3, particles move in circles under magnetic field B'; the radius reveals mass-to-charge ratio: r = mv/(qB').
Magnetic Field Due to a Moving Point Charge
• Magnetic field from a moving charge: B = (μ₀/4π)(qv × r̂)/r².
• Field lines form concentric circles around the path of motion (right-hand rule for direction).
• Field strength depends on speed, charge, and distance.
Biot-Savart Law: Magnetic Field from Current-Carrying Wire
• dB = (μ₀/4π)(Idl × r̂)/r², where dl is a small segment of wire and r is the distance to the field point.
• Magnetic fields are strongest near the wire and decrease with distance.
• Principle of superposition applies: total field is the vector sum of contributions from all segments.
Calculating Magnetic Fields with Biot-Savart Law
• Use symmetry to identify field direction and nonzero components.
• Express dl properly: dl = dx, dy, dz for straight wires; dl = r dθ for circular arcs.
• Integrate scalar components carefully over the wire geometry.
Example: Magnetic Field of an Infinite Straight Wire
• Field at distance r from a long straight wire: B = (μ₀I)/(2πr).
• Direction given by right-hand rule: thumb points current direction, fingers curl field direction.
Force Between Parallel Current-Carrying Wires
• Two parallel wires exert forces on each other.
• Force per unit length: F/L = (μ₀I₁I₂)/(2πr).
• Same direction currents attract; opposite direction currents repel.
Force on a Semicircular Wire in a Magnetic Field
• For a semicircle of radius a, net force is equivalent to that on a straight wire of length 2a.
• Force points according to the right-hand rule and depends on current and field strength.
Ampere’s Law
• Ampere’s law relates the line integral of magnetic field around a closed path to enclosed current: ∮B · dl = μ₀Ienclosed.
• Useful for symmetric cases like long straight wires, solenoids, and toroids.
• Choosing an appropriate Amperian path simplifies calculations greatly.
Using Ampere’s Law: Key Steps
• Pick a symmetric path (usually circle, rectangle, etc.) passing through the point of interest.
• Ensure B is constant along segments or zero where appropriate.
• Sum contributions ∮B · dl, setting B parallel or perpendicular to dl as needed.
• Solve for B using Ampere’s law once Ienclosed is known.
Example: Magnetic Field Outside and Inside a Wire
• Outside wire (r > b): B = (μ₀I)/(2πr).
• Inside wire (r < b) with uniform current: B ∝ r.
• At the wire’s surface (r = b), field is continuous and matches both expressions.
Magnetic Field from Non-Uniform Current Density
• If current density J varies with position, enclosed current must be calculated by integration.
• Example: For J(r) = ar (linear function of radius), the enclosed current inside radius R is Ienclosed = (2πaR³)/3.
• Apply Ampere’s law using calculated Ienclosed to find B inside and outside the wire.
Magnetic Field Inside a Solenoid
• A solenoid is a coil of wire with many closely spaced turns.
• Inside a long solenoid, the magnetic field is nearly uniform: B = μ₀nI, where n = turns per unit length.
• Outside an ideal solenoid, the magnetic field is approximately zero.
Magnetic Field Inside a Toroid
• A toroid is a solenoid bent into a circle.
• Magnetic field inside the torus (radius r): B = (μ₀NI)/(2πr), where N is the total number of turns.
• Field is confined inside the torus and depends inversely on r.
General Strategy for Applying Ampere’s Law
• Always choose a path where B is easy to evaluate (symmetry is crucial).
• Enclose as much current as possible without making B direction complicated.
• Remember that current into the page is negative, out of the page is positive according to the right-hand rule.
In a Nutshell
Magnetic fields originate from moving charges and electric currents. They exert forces on other moving charges, always perpendicular to velocity and field direction. The Biot-Savart law provides the magnetic field due to current segments, but Ampere’s law greatly simplifies calculations for symmetric cases. Understanding magnetic field behavior in wires, loops, solenoids, and toroids is fundamental for analyzing electromagnetic systems.