Magnetic Force on a Wire

Overview

This calculator simulates the magnetic force on a straight current-carrying wire placed in a uniform magnetic field. It uses the standard textbook relation

F = I · L · B · sin(θ)

and embeds that force in a simple mass–spring–damper model so you can visualize how the wire would move, how its energy changes, and how different parameters affect the motion.

Use the form above to set the current, wire length, magnetic field, angle, and mechanical parameters. The simulation panel then shows the wire deflection, while the energy bars and text summary describe what is happening over time.

Key variables and units

• I – Current through the wire, in amperes (A).
• L – Effective length of the wire within the magnetic field, in meters (m).
• B – Magnetic flux density (magnetic field strength), in tesla (T).
• θ – Angle between the current direction and the magnetic field, in degrees (°).
• m – Mass of the moving wire plus any attached mass, in kilograms (kg).
• k – Spring constant representing mechanical stiffness, in newtons per meter (N/m).
• c – Damping coefficient modeling friction or air resistance, in newton-seconds per meter (N·s/m).
• Δt – Numerical time step used in the simulation, in seconds (s).

All inputs are interpreted in SI units. If you are working with other unit systems, convert them to SI before entering values.

Magnetic force formula

For a straight conductor of length L, carrying current I in a uniform magnetic field of magnitude B, the magnitude of the magnetic force is

where θ is the angle between the current direction and the magnetic field. The direction of the force is perpendicular to both the current and the field and can be found using the right-hand rule.

Special cases:

• θ = 0° or 180° → sin(θ) = 0, so F = 0 (no magnetic force).
• θ = 90° → sin(θ) = 1, so F = I · L · B (maximum force).

Mass–spring–damper model

The calculator treats the wire as a rigid bar attached to a linear spring and damper, constrained to move in one vertical direction. The magnetic force acts as a constant driving force. The vertical displacement of the wire is denoted by y(t). The equation of motion is

Rearranging into the standard damped harmonic oscillator form gives

The right-hand side is a constant drive term, because the magnetic force does not depend on y in this simplified model.

Energy quantities

The simulation also tracks mechanical energy components:

• Kinetic energy: KE = ½ m v², where v is the wire's velocity.
• Spring potential energy: PE = ½ k y².
• Total mechanical energy: E = KE + PE, which decreases over time if damping is present.

Energy bars and numerical labels help you see how energy is exchanged between kinetic and potential forms and dissipated by damping.

Interpreting the results panel

After you enter inputs and start the simulation, the output area typically shows:

• A wire diagram that deflects up and down as the system evolves.
• A text summary indicating whether inputs are valid and how the system is behaving (for example, ready state or active simulation).
• Energy bars and labels for total, kinetic, and potential energy.
• (If enabled) a data download with time, displacement, velocity, and energy values for further analysis.

If your inputs are inconsistent (for example, missing values, negative mass, or non-numeric entries), the summary will indicate that the inputs are invalid and the simulation will not run. Adjust the form values and try again.

Worked example

Consider a wire with the following parameters:

• I = 10 A
• L = 0.20 m
• B = 0.50 T
• θ = 90°
• m = 0.050 kg
• k = 5.0 N/m
• c = 0.10 N·s/m

First compute the magnetic force:

sin(90°) = 1, so

F = I · L · B · sin(θ) = 10 × 0.20 × 0.50 × 1 = 1.0 N

This 1.0 N upward force acts against the spring and damping. Initially, if the wire starts from rest at y = 0, the acceleration at t = 0 is

a(0) = F / m = 1.0 / 0.050 = 20 m/s²

As the wire moves upward, the spring force k y grows and the motion slows, overshoots, and oscillates until the forces balance at a static deflection where

k y_eq ≈ F, so y_eq ≈ F / k = 1.0 / 5.0 = 0.20 m

Damping causes these oscillations to decay so the wire settles near the equilibrium displacement.

Comparison of special cases

Model assumptions and limitations

• Uniform magnetic field: The field B is taken as constant in space and time over the full wire length.
• Rigid straight wire: The wire does not bend or change shape; only its overall position changes.
• Single degree of freedom: Motion is restricted to one vertical direction, modeled as a mass attached to a linear spring and damper.
• Small to moderate deflections: The model remains linear; nonlinear effects such as large rotations or geometric stiffening are not included.
• No electrical losses: Heating, resistance changes, and inductive effects are ignored; the current I is assumed constant.
• 2D visualization: The true 3D direction of the Lorentz force is projected into the plane of the screen for clarity.
• Numerical integration limits: Very large time steps can make the numerical solution inaccurate or unstable; the calculator clamps Δt to a safe range.

Because of these simplifications, the tool is best understood as an educational visualization of F = I · L · B · sin(θ) and basic oscillations, not as a detailed design tool for precision electromagnetic machinery.

Using the calculator effectively

To explore the behavior of the system:

• Start with moderate default values for all inputs and run the simulation.
• Change one input at a time (for example, double the current or damping) to see how the motion and energies respond.
• Use the equilibrium estimate y_eq ≈ (I L B sin θ) / k as a quick check on whether the simulated deflection is reasonable.
• If the motion looks numerically unstable, reduce the time step Δt or the stiffness k and rerun.

This approach helps you build intuition about magnetic forces on conductors and the dynamics of simple mechanical systems.