Magnetic Force on a Wire

Introduction to magnetic force on a current-carrying wire

This calculator models the magnetic force on a straight wire carrying current through a uniform magnetic field. It applies the standard relation

F = I · L · B · sin(θ)

and uses that force as the constant drive in a simple mass–spring–damper simulation. This lets you observe the wire's modeled deflection, velocity, and mechanical energy while changing electrical, magnetic, and mechanical inputs.

Set the current, active wire length, field strength, angle, and mechanical parameters in the form. The simulation then displays the wire's vertical displacement and reports its time, displacement, velocity, and energy distribution as it evolves.

Magnetic-wire variables and SI units

For this current-carrying-wire force model, each input has the following meaning and SI unit:

• 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).

The magnetic-force calculation and the motion model use SI inputs. Convert any other units before entering values so that the reported force is in newtons and the displayed displacement is in meters.

Magnetic force formula for a straight wire

For a straight conductor of length L carrying current I through a uniform magnetic field of magnitude B, this wire calculator uses the force magnitude

Here, θ is the angle between the current direction and the magnetic field. The force direction is perpendicular to both directions; the animation represents that force as vertical motion in its simplified screen-plane model.

For the angle input, two useful magnetic-force cases are:

• 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 magnitude).

Mass–spring–damper model of wire deflection

The magnetic-wire simulation treats the wire as a rigid bar attached to a linear spring and damper and constrained to one vertical direction. Its magnetic force is constant for fixed inputs, and the vertical displacement is denoted by y(t). The equation used for the motion is

Dividing the magnetic-wire equation of motion by mass gives the form integrated by the simulation:

The right-hand side remains constant while current, length, field, and angle remain fixed. The spring term opposes displacement, while the damping term opposes velocity.

Mechanical energy in the wire simulation

The current-carrying-wire simulation tracks these 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.

The magnetic force can add mechanical energy as it drives the wire, while damping removes energy from the motion. The bars show the current total mechanical energy and the kinetic and spring-energy shares, rather than an energy balance that includes work supplied by the magnetic force.

Reading the magnetic-wire simulation results

When the magnetic-force simulation is initialized or playing, the output area shows:

• A wire diagram whose vertical position follows the modeled displacement.
• A text summary with the simulation time, displacement, and velocity.
• Energy bars and labels for total mechanical energy and its kinetic and spring components.
• A CSV download containing time, displacement, velocity, kinetic energy, and spring potential energy after simulation steps have been recorded.

If a required value is missing or non-numeric, or if mass or time step is not positive, the wire simulation reports invalid inputs and does not initialize. Nonnegative spring and damping values are accepted, including zero for an undamped or unstiffened idealized case.

Worked example: magnetic force and wire equilibrium

With the calculator's displayed default magnetic-wire inputs, the force calculation can be checked directly:

• 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

Because sin(90°) = 1, the magnetic force is:

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

Starting from rest at y = 0, the modeled spring and damping forces are initially zero. The initial acceleration is therefore:

a(0) = F / m = 1.0 / 0.050 = 20 m/s²

As the wire moves, spring force increases with displacement and damping resists its velocity. For this constant-force model, the static equilibrium condition is k y_eq = F, giving y_eq = F / k = 1.0 / 5.0 = 0.20 m. Damping causes transient oscillations about that position to decay.

Comparison of magnetic-force and motion cases

Changing magnetic or mechanical inputs affects the current-carrying wire model in different ways:

Magnetic-wire model assumptions and limitations

This magnetic-force animation makes the following simplifying assumptions:

• 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 assumptions, this tool is an educational visualization of magnetic force on a current-carrying conductor and its idealized spring-supported motion, not a detailed design model for electromagnetic machinery.

How to use the magnetic force on a wire calculator

To explore magnetic force and wire deflection systematically:

• Begin with the displayed magnetic, electrical, and mechanical values, then initialize or play the simulation.
• Change one input at a time—for example, increase current, rotate the field angle, or add damping—and compare the resulting motion.
• Use the equilibrium estimate y_eq = (I L B sin θ) / k as a check when the spring constant is positive.
• If the numerical motion appears unstable, choose a smaller time step Δt or reduce the stiffness k before running again.

Testing the current, field angle, stiffness, and damping separately helps distinguish what changes the magnetic driving force from what changes the wire's mechanical response.