Magnetic Field Calculator using Radius and Voltage
Calculate the magnetic field required to bend a charged particle’s path into a circle.
Particle Velocity: 5.93e+7 m/s
Magnetic Field vs. Radius (at constant Voltage)
What is a Magnetic Field using Radius and Voltage Calculator?
A **calculator magnetic field using radius and voltage** is a physics tool used to determine the strength of a magnetic field (B) required to force a moving charged particle into a circular path of a specific radius (r). This scenario is fundamental in many areas of physics and engineering, particularly in devices like mass spectrometers and particle accelerators. The “voltage” aspect refers to the accelerating potential difference that gives the charged particle its initial kinetic energy and velocity before it enters the magnetic field.
This calculator is essential for students, physicists, and engineers who need to design or analyze systems where charged particle beams are manipulated by magnetic fields. For example, it can help determine the necessary magnet strength to guide an electron beam in a cathode-ray tube or to separate isotopes in a mass spectrometer. Understanding this relationship is a key part of electromagnetism.
The Formula and Explanation
The operation of this **calculator magnetic field using radius and voltage** is based on two core principles of physics: the work-energy theorem and the Lorentz force acting as a centripetal force.
- Energy from Voltage: A charged particle with charge q gains kinetic energy (KE) when accelerated through a potential difference (voltage) V. This is given by:
KE = q * V
This kinetic energy is also equal to ½mv², where m is the mass and v is the velocity. We can therefore find the particle’s velocity:
v = √(2 * q * V / m) - Magnetic Force as Centripetal Force: When the particle enters a magnetic field B at a right angle, it experiences a magnetic (Lorentz) force, F = qvB. This force is always perpendicular to the velocity, providing the centripetal force needed for circular motion, F_c = mv²/r.
By equating these two forces (qvB = mv²/r) and solving for the magnetic field B, we get:
B = (m * v) / (q * r)
Substituting the expression for velocity (v) from the first step gives the final formula used by the calculator:
B = (1/r) * √(2 * m * V / q)
Variables Table
| Variable | Meaning | Unit (SI) | Typical Range |
|---|---|---|---|
| B | Magnetic Field Strength | Tesla (T) | 10⁻⁵ T – 10 T |
| r | Radius of Circular Path | meters (m) | 10⁻³ m – 10 m |
| V | Accelerating Voltage | Volts (V) | 1 V – 10⁷ V |
| m | Mass of the Particle | kilograms (kg) | 9.11×10⁻³¹ kg (electron) – 6.64×10⁻²⁷ kg (alpha) |
| q | Charge of the Particle | Coulombs (C) | 1.602×10⁻¹⁹ C (elementary charge) |
Practical Examples
Example 1: Electron Beam in an Old TV
An old cathode-ray tube (CRT) television accelerates electrons using a high voltage. Suppose we want to bend this electron beam into a circular path with a small radius for a focusing system.
- Inputs:
- Particle: Electron
- Accelerating Voltage (V): 25 kV (25,000 V)
- Radius (r): 2 cm (0.02 m)
- Results:
- Particle Velocity (v): ~9.38 x 10⁷ m/s (about 31% the speed of light)
- Required Magnetic Field (B): ~0.0267 Tesla
Example 2: Proton in a Cyclotron
A small-scale cyclotron accelerates protons. We need to find the magnetic field required to hold a proton in a specific orbit after being accelerated.
- Inputs:
- Particle: Proton
- Accelerating Voltage (V): 100 kV (100,000 V)
- Radius (r): 30 cm (0.3 m)
- Results:
- Particle Velocity (v): ~4.38 x 10⁶ m/s
- Required Magnetic Field (B): ~0.152 Tesla
How to Use This Magnetic Field Calculator
Using this **calculator magnetic field using radius and voltage** is straightforward. Follow these steps for an accurate result:
- Select the Particle Type: Choose ‘Electron’, ‘Proton’, or ‘Alpha Particle’ from the dropdown. This automatically sets the correct mass (m) and charge (q) for the calculation.
- Enter the Accelerating Voltage: Input the voltage value used to accelerate the particle. Use the dropdown to select the correct unit (Volts or Kilovolts). The tool will handle the conversion.
- Enter the Path Radius: Input the desired radius of the circular path the particle should follow inside the magnetic field. Select the appropriate unit (meters, centimeters, or millimeters).
- Interpret the Results: The calculator instantly updates. The primary result is the required magnetic field strength in Tesla (T). You can also see the intermediate values for the particle’s kinetic energy and final velocity, which are helpful for a deeper analysis. For more tools, you might be interested in a Lorentz Force Calculator.
Key Factors That Affect the Magnetic Field
Several factors influence the required magnetic field strength. Understanding them helps in predicting outcomes and designing experiments.
- Particle Mass (m): A heavier particle has more inertia, so a stronger magnetic field is needed to bend its path. B is proportional to √m.
- Particle Charge (q): The magnetic force is directly proportional to the charge. A particle with a higher charge experiences a greater force, so a weaker magnetic field is needed. B is proportional to 1/√q.
- Accelerating Voltage (V): A higher voltage imparts more kinetic energy and velocity to the particle. A faster particle is harder to bend, requiring a stronger magnetic field. B is proportional to √V.
- Path Radius (r): To achieve a smaller, tighter circle, you need to apply a much stronger force. Therefore, the required magnetic field is inversely proportional to the radius. B is proportional to 1/r.
- Particle Velocity (v): This is an intermediate factor determined by V, m, and q. As velocity increases, the required magnetic field increases linearly (if you could set velocity directly). Check out our Peak Voltage Calculator for related calculations.
- Field Direction: The formula assumes the particle enters the magnetic field perpendicularly. If it enters at an angle, the motion becomes helical (a spiral), and this calculator for pure circular motion would not be fully applicable.
Frequently Asked Questions (FAQ)
- What is the unit of a magnetic field?
- The standard SI unit for magnetic field strength (or magnetic flux density) is the Tesla (T). A smaller unit, the Gauss (G), is also used, where 1 T = 10,000 G.
- Why does the particle’s energy not change in the magnetic field?
- The magnetic force is always perpendicular to the particle’s direction of motion. Because work is done only when force has a component parallel to displacement, the magnetic field does no work on the particle. Thus, its kinetic energy and speed remain constant; only its direction changes. This is why a Voltage Divider Calculator is a different concept entirely.
- What happens if the voltage is zero?
- If the voltage is zero, the particle has no initial velocity. It will not move and therefore will not experience any magnetic force. The required magnetic field would be irrelevant.
- Can I use this for any charged particle?
- Yes, in principle. The calculator is pre-set for electrons, protons, and alpha particles. For other ions, you would need to know their specific mass and charge and use the formula directly.
- Does this calculator account for relativistic effects?
- No. At very high voltages (approaching the mega-volt range), a particle’s speed can become a significant fraction of the speed of light. This calculator uses classical mechanics (v = √(2qV/m)), which does not account for relativistic mass increase. For highly relativistic particles, the required magnetic field would be greater than predicted here.
- How do I change the units for the inputs?
- Next to the input fields for voltage and radius, there are dropdown menus. You can select different units (e.g., kV for voltage, cm for radius), and the **calculator magnetic field using radius and voltage** will automatically use the correct conversion factor.
- What is a mass spectrometer?
- A mass spectrometer is a device that uses this exact principle. It ionizes atoms, accelerates them with a voltage, and passes them through a magnetic field. Particles of different masses will curve with different radii, allowing the device to separate and identify them. You can learn more about Faraday’s Law which is also related to electromagnetism.
- What if the particle’s velocity is not perpendicular to the magnetic field?
- If there is a component of velocity parallel to the magnetic field, that component will be unaffected. The particle will still move in a circle in the perpendicular plane, but it will also drift along the field lines, resulting in a helical (corkscrew-shaped) path.
Related Tools and Internal Resources
If you found this tool useful, explore our other engineering and physics calculators:
- Lorentz Force Calculator: Calculate the force on a charge moving in electric and magnetic fields.
- Voltage Divider Calculator: A tool for designing circuits to produce a specific output voltage.
- Lenz’s Law and Faraday’s Law Calculator: Explore the principles of electromagnetic induction.
- Peak Voltage Calculator: Convert between RMS, peak, and average AC voltage values.