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How to Find Magnetic Field Using Velocity?

Published in Electromagnetism 4 mins read

To find the magnetic field using velocity, you'll typically use the relationship between magnetic force, charge, velocity, and the magnetic field itself. The specific formula and approach depend on the information you have available.

Here's a breakdown of common scenarios and formulas:

1. Knowing the Magnetic Force, Charge, and Velocity:

The fundamental equation linking these quantities is:

F = q(v x B)

Where:

  • F is the magnetic force vector (in Newtons, N)
  • q is the charge of the particle (in Coulombs, C)
  • v is the velocity vector of the charge (in meters per second, m/s)
  • B is the magnetic field vector (in Teslas, T)
  • (v x B) is the cross product of the velocity and magnetic field vectors.

To find B, you'd need to:

  1. Determine the direction of the force: The force is perpendicular to both the velocity and the magnetic field. The right-hand rule is commonly used to determine the direction of the force, velocity, and magnetic field relative to each other. Point your index finger in the direction of the velocity, your middle finger in the direction of the magnetic field. Your thumb will then point in the direction of the force on a positive charge.

  2. Calculate the magnitude: Since F = qvBsin(θ), where θ is the angle between v and B, the magnitude of the magnetic field can be found if the angle between the velocity and magnetic field is known. The equation becomes:

    B = F / (qv sin(θ))

    If v and B are perpendicular (θ = 90°), then sin(θ) = 1, simplifying the equation to B = F / qv.

Example:

Suppose a particle with a charge of 2 C is moving at 3 m/s perpendicular to a magnetic field. The force on the particle is 12 N.

B = 12 N / (2 C 3 m/s sin(90°)) = 12 N / 6 = 2 T.

2. Knowing the Radius of Circular Motion, Mass, Charge, and Velocity:

If a charged particle moves perpendicular to a uniform magnetic field, it will experience a force that causes it to move in a circle. The radius (r) of this circle is given by:

r = mv / (qB)

Where:

  • r is the radius of the circular path (in meters, m)
  • m is the mass of the particle (in kilograms, kg)
  • v is the speed of the particle (in meters per second, m/s)
  • q is the charge of the particle (in Coulombs, C)
  • B is the magnetic field strength (in Teslas, T)

To find B, rearrange the equation:

B = mv / (qr)

Example:

An electron (mass = 9.11 x 10-31 kg, charge = 1.602 x 10-19 C) is moving at 1 x 107 m/s in a circular path of radius 0.02 m in a magnetic field.

B = (9.11 x 10-31 kg 1 x 107 m/s) / (1.602 x 10-19 C 0.02 m) ≈ 0.00284 T

3. Using Voltage and Radius:

The reference mentions r=mVqB where V is voltage, but this is not a standard physics equation. It appears to be a simplified or derived equation specific to a particular context. A more standard approach when kinetic energy (related to voltage) and radius are involved is to use the following:

  1. Kinetic Energy: KE = (1/2)mv2 = qV, where V is the accelerating potential (voltage). This allows you to relate the voltage to the velocity of the particle.
  2. Radius of Circular Motion: r = mv/qB (as discussed above).

To find B:

  1. Solve for v from the kinetic energy equation: v = sqrt(2qV/m)
  2. Substitute this expression for v into the radius equation: r = m * sqrt(2qV/m) / (qB)
  3. Rearrange to solve for B: B = (m * sqrt(2qV/m)) / (qr) = sqrt(2mV) / (qr)

Important Considerations:

  • Units: Always ensure you're using consistent units (SI units are generally preferred).
  • Direction: Remember that magnetic fields are vector quantities. You'll need to determine both the magnitude and direction. Use the right-hand rule.
  • Uniformity: The formulas above typically assume a uniform magnetic field.

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