Moving Charges And Magnetism Formulas

• Lorentz force on a charge (q) moving with velocity (v) in a magnetic field (B) [As vector cross product]:

$$\mathbf{F}_B = q (\mathbf{v} \times \mathbf{B})$$

• Magnitude of Lorentz force

$$F_B = q v B \sin\theta$$

• Lorentz Force when both the Electric and Magnetic fields are acting on the charge

$$\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$$

• Force on a Straight conductor of length (L) carrying a current (I) in a magnetic field (B)

$$\mathbf{F} = I (\mathbf{L} \times \mathbf{B})$$

• Magnitude of force

$$F=ILB\sin \theta$$

• Force when the conductor is parallel to B:

$$F=0$$

• Force when the conductor is perpendicular to B:

$$F=ILB$$

• Path of Motion
  • When velocity is perpendicular to the magnetic field, it revolves in a circular path.
  • When the velocity vector is parallel to the magnetic field, it revolves in a straight line.
  • If the velocity vector has both components, then it revolves in a helical path.

• Radius of Circular Path

  $$r = \frac{m v_\perp}{q B}$$

   

• Angular Frequency (Cyclotron Frequency)

  $$\omega = \frac{q B}{m}$$

   

• Time Period

  $$T = \frac{2\pi m}{q B}$$

   

• Pitch of Helix

  $$p=v_{<br>}\times T=\frac{2\pi m{v}_{}}{qB}$$

• Magnetic Field dB due to a Current-carrying conductor due to current element dI, at a distance r:

$$d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \hat{\mathbf{r}}}{r^2}$$

• Magnitude of the Magnetic field:

$$dB = \frac{\mu_0}{4\pi} \frac{I \, dl \sin\theta}{r^2}$$

• Magnetic Field on the Axis of a Circular Current Loop

$$B=\frac{{\mu }_{0}I{R}^2}{2(R^2+x^2{)}^{3\mathrm{/}2}}$$

• At the center of the loop (x $=0$):

$$B_{\text{center}}=\frac{{\mu }_{0}I}{2R}$$

• Circuital Law Formula

$$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enclosed}}$$

Applications of Ampere's Circuital Law

• Magnetic Field Due to a Long Straight Current-Carrying Wire of Infinite Length

• Magnetic Field Inside a Long Solenoid with n turns per unit length

$$B = \mu_0 n I$$

• Inside a Toroid of radius r with N turns 

$$B=\frac{{\mu }_{0}NI}{2\pi r}$$

• Magnetic Field Inside a Long Solenoid

$$B = \mu_0 n I$$

• Magnetic Field Inside a Solenoid with Core

$$B=\mu nI$$

• Magnetic Field Outside a Long Solenoid

$$B\approx 0$$

• Magnetic Flux Through Solenoid

$$\mathrm{\Phi }=B\cdot A=({\mu }_{0}nI)\cdot A$$

• Force between Two Parallel Current-Carrying Conductors

$$\frac{F}{L}=\frac{{\mu }_0{I}_1{I}_2}{2\pi d}$$

• Torque on a Current Loop in a Magnetic Field

• Magnitude:

$$\tau =mB\sin \theta$$

• Magnetic Dipole Moment

$$\mathbf{m}=nIA\hat{n}$$

• Magnitude:

$$m=nIA$$

• Deflecting Torque: $\tau_d = N \cdot I \cdot A \cdot B$
  
• Restoring Torque: $\tau_r = k \cdot \theta$
  
• Reading at Equilibrium: $I = \frac{k \cdot \theta}{N \cdot A \cdot B}$
  
• Current Sensitivity: $S = \frac{\theta}{I} = \frac{N \cdot A \cdot B}{k}$
  
• Voltage Sensitivity: $S_v = \frac{\theta}{V} = \frac{N \cdot A \cdot B}{k \cdot R}$