Moving Charges & Magnetism — Complete Formula

Magnetic Force

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Force on Charge	[\vec{F} = q \vec{v} \times \vec{B}]	[q] = charge, [v] = velocity, [B] = magnetic field	N	Perpendicular force ⭐
Magnitude	[F = qvB\sin\theta]	[\theta] = angle between [v] and [B]	N	Zero if parallel ⚠️

💡 Memory Hint:
Magnetic force → always perpendicular

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Radius	[r = \dfrac{mv}{qB}]	[m] = mass	m	Circular motion ⭐
Time Period	[T = \dfrac{2\pi m}{qB}]	—	s	Independent of speed ⭐
Frequency	[f = \dfrac{qB}{2\pi m}]	—	Hz	Cyclotron frequency
Pitch (Helical)	[p = v_\parallel T]	[v_\parallel] = parallel component	m	Spiral motion

💡 Memory Hint:
Magnetic field changes direction, not speed

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Force	[\vec{F} = I \vec{L} \times \vec{B}]	[I] = current, [L] = length vector	N	Fleming’s left-hand rule ⭐
Magnitude	[F = BIL\sin\theta]	—	N	Max when perpendicular ⭐

💡 Memory Hint:
Use left-hand rule for direction

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Torque	[\tau = NIAB\sin\theta]	[N] = turns, [A] = area	N·m	Rotates loop ⭐
Magnetic Moment	[m = NIA]	[m] = magnetic moment	A·m²	Important ⭐
Torque Form	[\tau = mB\sin\theta]	—	N·m	Simplified form

💡 Memory Hint:
Loop behaves like a magnetic dipole

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Biot–Savart Law	[dB = \dfrac{\mu_0}{4\pi} \dfrac{Idl\sin\theta}{r^2}]	[\mu_0] = permeability	T	Basic law ⭐

💡 Memory Hint:
Current element produces magnetic field

System	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Straight Wire	[B = \dfrac{\mu_0 I}{2\pi r}]	[r] = distance	T	Circular field ⭐
Circular Loop (center)	[B = \dfrac{\mu_0 I}{2R}]	[R] = radius	T	Important ⭐
Solenoid	[B = \mu_0 n I]	[n] = turns per length	T	Uniform field ⭐
Toroid	[B = \dfrac{\mu_0 NI}{2\pi r}]	—	T	Inside only ⚠️

💡 Memory Hint:
More current → stronger magnetic field

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Ampere Law	[\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{enc}]	[I_{enc}] = enclosed current	—	Used in symmetry ⭐

💡 Memory Hint:
Magnetic field depends on enclosed current

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Force per Length	[\dfrac{F}{L} = \dfrac{\mu_0 I_1 I_2}{2\pi r}]	—	N/m	Defines ampere ⭐
Nature	Attraction/Repulsion	—	—	Same → attract ⭐

💡 Memory Hint:
Same direction → attract

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Dipole Moment	[M = m \cdot 2l]	[m] = pole strength	A·m²	Analog of electric dipole ⭐
Torque	[\tau = MB\sin\theta]	—	N·m	Aligns with field
Potential Energy	[U = -MB\cos\theta]	—	J	Stable when aligned ⭐

💡 Memory Hint:
Magnet tries to align with field

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Current Sensitivity	[S_i = \dfrac{\theta}{I}]	[\theta] = deflection	—	Higher → more sensitive
Voltage Sensitivity	[S_v = \dfrac{\theta}{V}]	—	—	Depends on resistance

💡 Memory Hint:
Galvanometer → detects small current

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Ammeter	[S = \dfrac{IG}{I – IG}]	[S] = shunt resistance	Ω	Parallel connection ⭐
Voltmeter	[R = \dfrac{V}{IG} – G]	[G] = galvanometer resistance	Ω	Series connection ⭐

💡 Memory Hint:

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