# Magnetic force equation

The magnetic force equation, derived from the Lorentz force, calculates the magnetic force (F) acting on a charged particle. For a moving charge, the equation is F = q v B sinθ, where q represents the charge, v is the velocity, B is the magnetic field, and θ is the angle between the velocity vector and the magnetic field vector.

For a current-carrying wire, the magnetic force equation is expressed as F = I L B sinθ, where I denotes the electric current flowing through the wire, L represents the length of the wire, B is the magnetic field, and θ is the angle between the wire and the direction of magnetic field.

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## Practice problems

### Problem #1

Find the magnetic force acting on a particle with a charge of 6 µC as it moves perpendicular to a magnetic field with a strength of 0.5 T at a velocity of 1.5 × 106 m/s.

Solution

Given data:

• Magnetic force acting on a particle, F = ?
• Charge of a moving particle, q = 6 µC = 6 × 10-6 C
• Angle between velocity vector and magnetic field vector, θ = 90°
• Magnetic field, B = 0.5 T
• Velocity of a particle, v = 1.5 × 106 m/s

Applying the formula, for a moving charge:

• F = q v B sinθ
• F = 6 × 10-6 × 1.5 × 106 × sin (90°)
• F = 6 × 1.5 × 1
• F = 9 N

Therefore, the magnetic force acting on a particle is 9 N.

### Problem #2

Determine the magnetic force acting on a 15 cm wire with a current of 4 A that is oriented at a 45° angle from parallel to a magnetic field with a strength of 5 T.

Solution

Given data:

• Magnetic force acting on a wire, F = ?
• Length of a wire, L = 15 cm = 0.15 m
• Electric current, I = 4 A
• Wire is oriented from parallel at an angle, θ = 45°
• Magnetic field, B = 5 T

Applying the formula, for a current-carrying wire:

• F = I L B sinθ
• F = 4 × 0.15 × 5 × sin (45°)
• F = 3 × 0.7071
• F = 2.12 N

Therefore, the magnetic force acting on a wire is 2.12 N.

### Problem #3

Find the magnetic force experienced by a 7 µC charge moving at 5.1 × 107 m/s through a magnetic field with a strength of 10 T at an angle of 36° from perpendicular to the magnetic field.

Solution

Given data:

• Magnetic force acting on a particle, F = ?
• Charge of a moving particle, q = 7 µC = 7 × 10-6 C
• Velocity of a particle, v = 5.1 × 107 m/s
• Magnetic field, B = 10 T
• Charge moves from perpendicular at an angle, θ = 36°

Applying the formula, for a moving charge:

• F = q v B sinθ
• F = 7 × 10-6 × 5.1 × 107 × sin (36°)
• F = 7 × 5.1 × 10 × 0.5877
• F = 209.80 N

Therefore, a moving charge experiences the magnetic force of 209.80 N.

### Problem #4

Calculate the magnetic force acting on a 10 cm wire with a current of 9 A that is oriented at a 25° angle from parallel to a magnetic field with a strength of 12 T.

Solution

Given data:

• Magnetic force acting on a wire, F = ?
• Length of a wire, L = 10 cm = 0.1 m
• Electric current, I = 9 A
• Wire is oriented from parallel at an angle, θ = 25°
• Magnetic field, B = 12 T

Applying the formula, for a current-carrying wire:

• F = I L B sinθ
• F = 9 × 0.1 × 12 × sin (25°)
• F = 10.8 × 0.4226
• F = 4.56 N

Therefore, the magnetic force acting on a wire is 4.56 N.