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PhysicsMotion in Magnetic Field
An electron (mass 9x10^-31 kg, charge 1.6x10^-19 C) moving with speed c/100 is injected into a magnetic field B of magnitude 9x10^-4 T perpendicular to its direction. To prevent deflection, a uniform electric field E is applied together with B. Then (c = 3x10^8 m/s):
Options
1
E is parallel to B, magnitude 27x10^4 V/m
2
E is perpendicular to B, magnitude 27x10^4 V/m
3
E is perpendicular to B, magnitude 27x10^2 V/m
4
E is parallel to B, magnitude 27x10^2 V/m
Correct Answer
E is perpendicular to B, magnitude 27x10^2 V/m
Solution
1

For no deflection: Electric force = Magnetic force

qE = qvB

E = vB

2

v = c/100 = 3x10^8/100 = 3x10^6 m/s

E = vB = (3x10^6)(9x10^-4) = 27x10^2 V/m = 2700 V/m

E must be perpendicular to B (and to v) to oppose the magnetic force.

E = vB = (3x10^6)(9x10^-4) = 27x10^2 V/m
E perpendicular to B | For no deflection: eE = evB
Theory: Motion in Magnetic Field
1. Velocity Selector Principle

A velocity selector has perpendicular E and B fields. For a charged particle moving perpendicular to both fields: Electric force F_E = qE (along E direction). Magnetic force F_B = qvB (perpendicular to v and B). For no deflection: qE = qvB, so v = E/B. Only particles with this specific velocity pass through undeflected. Used in mass spectrometers to select monoenergetic beams.

2. Lorentz Force

Total force on a charged particle: F = q(E + v x B). Electric force: qE (along E, independent of velocity). Magnetic force: q(v x B) (perpendicular to both v and B, depends on velocity). Magnetic force does no work (always perpendicular to displacement). Magnetic force cannot change speed, only direction. For circular motion in B: qvB = mv^2/r, so r = mv/qB.

3. Circular Motion in Magnetic Field

Charged particle moving perpendicular to uniform B: magnetic force provides centripetal force. qvB = mv^2/r. Radius r = mv/(qB). Period T = 2pi m/(qB) (independent of speed! - cyclotron principle). Frequency (cyclotron frequency) f = qB/(2pi m). Applications: cyclotrons, mass spectrometers, bubble chambers for particle detection.

4. Mass Spectrometer

Velocity selector selects v = E/B. Then particles enter magnetic field B and move in semicircle of radius r = mv/(qB). Mass m = qBr/v = qB^2 r/E. By measuring radius r, mass is determined. Used to: identify isotopes, measure atomic masses precisely, analyze chemical composition (analytical chemistry). Nobel Prize: J.J. Thomson (1906) discovered electron using this principle.

5. Cyclotron

Cyclotron accelerates charged particles using alternating electric field. Two D-shaped electrodes (dees) in uniform magnetic field. Particle spirals outward, gaining energy at each gap crossing. Cyclotron resonance condition: frequency of alternating field = cyclotron frequency = qB/2pi m. Maximum energy limited by relativistic mass increase (particles slow down relative to classical prediction). Synchrocyclotron and synchrotron overcome this by varying frequency or B.

6. Hall Effect

When current-carrying conductor is placed in perpendicular magnetic field: charge carriers deflect to one side. Creates transverse voltage (Hall voltage): VH = IB/(nqt). n = charge carrier density, t = thickness. Used to determine carrier type (electrons or holes) and carrier concentration. Hall sensors measure magnetic fields. Applications: position sensors in brushless DC motors, automotive sensors, current clamps.

7. Magnetic Force on Current

Force on current-carrying wire in magnetic field: F = IL x B. Magnitude: F = BIL sin(theta). Maximum when wire perpendicular to B. Zero when wire parallel to B. Two parallel wires carrying current: if same direction, attract. If opposite, repel. Force per unit length: F/L = mu0 I1 I2 / (2pi d). Definition of Ampere based on this force.

8. Motion of Charged Particles

Particle enters B at angle theta to field: component along B (v cos theta) is unaffected (no force). Component perpendicular to B (v sin theta) causes circular motion. Net result: helical motion with pitch = (v cos theta)(2pi m/qB). Particle spirals along field lines. This explains aurora borealis: solar wind particles spiral along Earth magnetic field lines and hit atmosphere at poles.

Frequently Asked Questions
1. Why is E perpendicular to B and not parallel? ⌄
In velocity selector, magnetic force F_B = q(v x B) is perpendicular to both v and B. To cancel this force, E must act in the opposite direction to F_B. Since F_B is perpendicular to B, E must also be perpendicular to B. If E were parallel to B, it would not oppose the magnetic force at all (force components in different directions cannot cancel). So E perpendicular to B is the necessary condition.
2. How does a mass spectrometer work using this principle? ⌄
A mass spectrometer has three stages: (1) Ion source: creates charged particles (ions). (2) Velocity selector: E and B fields crossed, selects only ions with v = E/B. (3) Magnetic analyzer: another B field bends the monovelocity ions into semicircles. Radius r = mv/qB. Heavier ions have larger r. Detector records where different masses hit. Mass = qBr/v = qB^2r/E. This way different isotopes of the same element (same q but different m) are separated and identified.
3. What is the cyclotron frequency? ⌄
Cyclotron frequency = qB/(2pi m). This is the frequency of circular motion of a charged particle in magnetic field B. Key property: independent of speed (as long as non-relativistic). This allows cyclotrons to work: as particle gains energy and speed, its orbit radius increases but frequency stays constant. The alternating accelerating voltage is tuned to this fixed frequency. For proton in B = 1 T: f = (1.6x10^-19 x 1)/(2pi x 1.67x10^-27) = 15.2 MHz.
4. Can magnetic force do work on a moving charge? ⌄
No. Magnetic force is always perpendicular to velocity (F = qv x B is perpendicular to v). Work = F.ds = F.v dt = 0 (since F perpendicular to v). Therefore magnetic force never changes the kinetic energy (speed) of a particle. It only changes the direction. This is why in cyclotrons, the acceleration (energy gain) comes from the electric field at the gap between dees, not from the magnetic field.
5. What happens when both E and B forces balance? ⌄
When eE = evB, net electromagnetic force on electron is zero. Electron travels in straight line (no deflection). This is the operating principle of a velocity selector: it selects a specific velocity v = E/B from a beam containing particles of various speeds. Only particles with exactly v = E/B pass through undeflected. Faster: deflected by B more than repelled by E. Slower: deflected by E more than by B.
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