Moving Charges and Magnetism — JEE Mains Physics

Formulas (2)

• Field of a long straight wire and of two parallel wires · Long straight wire
  $$B = \frac{\mu_0 I}{2\pi d}$$
• Field of a finite straight wire and of a polygon loop · Finite straight wire
  $$B = \frac{\mu_0 I}{4\pi d}\left(\sin\alpha_1 + \sin\alpha_2\right) \qquad dB = \frac{\mu_0}{4\pi}\frac{I\,dl\sin\theta}{r^{2}}$$

Reference tables (1)

Watch out for (9)

• Opposite currents add between the wires
  → Field of a long straight wire and of two parallel wires
• Outside the pair the rule reverses
  → Field of a long straight wire and of two parallel wires
• Fields at an angle add as vectors
  → Field of a long straight wire and of two parallel wires
• The angles are measured from the perpendicular
  → Field of a finite straight wire and of a polygon loop
• A semi-infinite wire gives half, not the full value
  → Field of a finite straight wire and of a polygon loop
• The centre of a polygon is not at a distance a/2
  → Field of a finite straight wire and of a polygon loop
• A straight piece aimed at the centre gives nothing
  → Field at the centre of arcs and bent wires
• The angle of an arc must be in radians
  → Field at the centre of arcs and bent wires
• Check the sense of every piece before adding
  → Field at the centre of arcs and bent wires

Formulas (2)

• Field at the centre of a circular coil · Centre of a coil and energy density
  $$B = \frac{\mu_0 N I}{2R} \qquad u = \frac{B^{2}}{2\mu_0}$$
• Field on the axis of a circular loop · Axis of a loop
  $$B = \frac{\mu_0 N I R^{2}}{2\left(R^{2} + x^{2}\right)^{3/2}} \qquad \frac{B}{B_c} = \frac{R^{3}}{\left(R^{2} + x^{2}\right)^{3/2}}$$

Watch out for (6)

• Rewinding changes the radius too
  → Field at the centre of a circular coil
• Perpendicular coils add as vectors
  → Field at the centre of a circular coil
• Do not forget the number of turns
  → Field at the centre of a circular coil
• The power is 3/2, not 1/2
  → Field on the axis of a circular loop
• An approximation needs x much less than R
  → Field on the axis of a circular loop
• The direction of each coaxial loop's field
  → Field on the axis of a circular loop

Formulas (2)

• Ampère's law for a solid wire, a hollow tube and a coaxial cable · Ampère's law and a solid wire
  $$\oint \vec B \cdot d\vec l = \mu_0 I_{\text{enc}} \qquad B_{\text{in}} = \frac{\mu_0 I r}{2\pi a^{2}},\quad B_{\text{out}} = \frac{\mu_0 I}{2\pi r}$$
• Field inside a solenoid and a toroid · Solenoid, intensity, core and toroid
  $$B = \mu_0 n I \qquad H = nI \qquad B = \mu_0\mu_r n I \qquad B_{\text{toroid}} = \frac{\mu_0 N I}{2\pi r}$$

Watch out for (6)

• Inside a solid wire B grows with r
  → Ampère's law for a solid wire, a hollow tube and a coaxial cable
• The field outside a coaxial cable is zero
  → Ampère's law for a solid wire, a hollow tube and a coaxial cable
• Only enclosed current counts
  → Ampère's law for a solid wire, a hollow tube and a coaxial cable
• Turns per centimetre must become turns per metre
  → Field inside a solenoid and a toroid
• H has no μ₀ in it
  → Field inside a solenoid and a toroid
• Flux and flux linkage differ by N
  → Field inside a solenoid and a toroid

Formulas (2)

• Magnetic force on a moving charge, F = q v × B · Magnetic force
  $$\vec F = q\left(\vec v \times \vec B\right), \qquad F = qvB\sin\theta$$
• Electric and magnetic forces together: the velocity selector · Lorentz force and the selected speed
  $$\vec F = q\left(\vec E + \vec v \times \vec B\right) \qquad v = \frac{E}{B}$$

Watch out for (6)

• v × B is not B × v
  → Magnetic force on a moving charge, F = q v × B
• An electron's force is reversed
  → Magnetic force on a moving charge, F = q v × B
• The magnetic force never changes speed
  → Magnetic force on a moving charge, F = q v × B
• The selector picks a speed, not a charge or a mass
  → Electric and magnetic forces together: the velocity selector
• Find v from the kinetic energy first
  → Electric and magnetic forces together: the velocity selector
• Constant velocity is impossible with only an electric field
  → Electric and magnetic forces together: the velocity selector

Formulas (3)

• Radius of a charge's circular path and how it compares between particles · Radius of the circle
  $$r = \frac{mv}{qB} = \frac{p}{qB} = \frac{\sqrt{2mK}}{qB} = \frac{1}{B}\sqrt{\frac{2mV}{q}}$$
• Finding a radius, a mass or a field from r = mv/qB · Mass from a measured radius
  $$m = \frac{qBr}{v} = \frac{qB^{2}r^{2}}{2V}$$
• Period of revolution, the helix and the cyclotron · Period, pitch and cyclotron energy
  $$T = \frac{2\pi m}{qB} \qquad p = v\cos\theta\,T \qquad K_{\max} = \frac{q^{2}B^{2}R^{2}}{2m}$$

Watch out for (9)

• Equal energy and equal voltage are different conditions
  → Radius of a charge's circular path and how it compares between particles
• Curvature is not radius
  → Radius of a charge's circular path and how it compares between particles
• Radius goes as the square root of kinetic energy
  → Radius of a charge's circular path and how it compares between particles
• Energy in eV must be turned into joules
  → Finding a radius, a mass or a field from r = mv/qB
• Radius, not diameter
  → Finding a radius, a mass or a field from r = mv/qB
• Singly ionised means charge e
  → Finding a radius, a mass or a field from r = mv/qB
• Only the part of v along B makes the pitch
  → Period of revolution, the helix and the cyclotron
• Two gains of energy per revolution in a cyclotron
  → Period of revolution, the helix and the cyclotron
• The period does not depend on the speed
  → Period of revolution, the helix and the cyclotron

Formulas (3)

• Force on a current-carrying wire in a magnetic field · Force on a wire
  $$\vec F = I\,\vec L \times \vec B, \qquad F = ILB\sin\theta$$
• Force between two parallel currents · Force per metre between parallel wires
  $$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$
• Magnetic moment of a coil and the torque on it · Moment and torque
  $$\vec m = NI\vec A, \qquad \vec \tau = \vec m \times \vec B, \qquad \tau = NIAB\sin\theta$$

Watch out for (9)

• Only the part in the field counts
  → Force on a current-carrying wire in a magnetic field
• A bent wire's force uses the chord
  → Force on a current-carrying wire in a magnetic field
• θ is the angle between the wire and the field
  → Force on a current-carrying wire in a magnetic field
• Like currents attract
  → Force between two parallel currents
• The two forces are equal even if the currents differ
  → Force between two parallel currents
• The force goes as the product, not the sum
  → Force between two parallel currents
• θ is measured from the axis, not the plane
  → Magnetic moment of a coil and the torque on it
• Opposite currents give opposite moments
  → Magnetic moment of a coil and the torque on it
• Do not forget N
  → Magnetic moment of a coil and the torque on it

Formulas (2)

• Moving coil galvanometer: deflection and sensitivity · Balance and sensitivity
  $$NIAB = C\theta \qquad \frac{\theta}{I} = \frac{NAB}{C} \qquad \frac{\theta}{V} = \frac{NAB}{CR}$$
• Converting a galvanometer into an ammeter with a shunt · Shunt
  $$S = \frac{I_g G}{I - I_g} \qquad R_A = \frac{SG}{S + G}$$

Reference tables (1)

Watch out for (9)

• More turns also mean more resistance
  → Moving coil galvanometer: deflection and sensitivity
• Figure of merit is the inverse of sensitivity
  → Moving coil galvanometer: deflection and sensitivity
• Current and voltage sensitivity differ by R
  → Moving coil galvanometer: deflection and sensitivity
• The shunt carries I − Ig, not I
  → Converting a galvanometer into an ammeter with a shunt
• A shunt goes in parallel
  → Converting a galvanometer into an ammeter with a shunt
• The ammeter's resistance is less than the shunt
  → Converting a galvanometer into an ammeter with a shunt
• Subtract the galvanometer's own resistance
  → Converting a galvanometer into a voltmeter, compared with an ammeter
• Raising a voltmeter's range uses the meter's resistance
  → Converting a galvanometer into a voltmeter, compared with an ammeter
• A voltmeter in series reads almost the whole supply
  → Converting a galvanometer into a voltmeter, compared with an ammeter