6.5.4 Force on a Current-Carrying Conductor

• The magnetic flux density B is defined as:

The force acting per unit current per unit length on a current-carrying conductor placed perpendicular to the magnetic field

• Rearranging the equation for magnetic force on a wire, the magnetic flux density is defined by the equation:

• Where:
  • B = magnetic flux density (T)
  • F = magnetic force (N)
  • I = current (A)
  • L = length of the wire (m)

• Note: this equation is only relevant when the B-field is perpendicular to the current
• Magnetic flux density is measured in units of tesla, which is defined as:

A wire carrying a current of 1 A normal to a magnetic field of flux density of 1 T with force per unit length of the conductor of 1 N m⁻¹

• To put this into perspective, the Earth's magnetic flux density is around 0.032 mT and an ordinary fridge magnet is around 5 mT
• The magnetic flux density is sometimes referred to as the magnetic field strength

Step 1: Write out the known quantities

• • Force on wire, F = 0.04 N
  • Current, I = 3.0 A
  • Length of wire, L = 15 cm = 15 × 10⁻² m

Step 2: Write out the magnetic flux density B equation

Step 3: Substitute in values

Step 4: Determine the direction of the B field

• • Using Fleming’s left-hand rule :

F = to the left

I = vertically upwards

therefore, B = into the page

• A current-carrying conductor produces its own magnetic field
  • An external magnetic field will therefore exert a magnetic force on it

• A current-carrying conductor (e.g. a wire) will experience the maximum magnetic force if the current through it is perpendicular to the direction of the magnetic flux lines
  • A simple situation would be a copper rod placed within a uniform magnetic field
  • When current is passed through the copper rod, it experiences a force which makes it accelerate

A copper rod moves within a magnetic field when current is passed through it

• The force F on a conductor carrying current I in a magnetic field with flux density B is defined by the equation

F = BIL sin θ

• Where:
  • F = magnetic force on the current-carrying conductor (N)
  • B = magnetic flux density of external magnetic field (T)
  • I = current in the conductor (A)
  • L = length of the conductor in the field (m)
  • θ = angle between the conductor and external flux lines (degrees)

• This equation shows that the magnitude of the magnetic force F is proportional to:
  • Current I 
  • Magnetic flux density B
  • Length of conductor in the field L
  • The sine of the angle θ between the conductor and the magnetic flux lines

Magnitude of the force on a current carrying conductor depends on the angle of the conductor to the external B field

• The maximum force occurs when sin θ = 1
  • This means θ = 90^o and the conductor is perpendicular to the B field
  • This equation for the magnetic force now becomes:

F = BIL

• The minimum force (0) is when sin θ = 0
  • This means θ = 0^o and the conductor is parallel to the B field

• It is important to note that a current-carrying conductor will experience no force if the current in the conductor is parallel to the field

Step 1: Write down the known quantities

• • Magnetic flux density, B = 80 mT = 80 × 10⁻³ T
  • Current, I = 0.87 A
  • Length of wire, L = 1.4 m
  • Angle between the wire and the magnetic flux lines, θ = 30^o

Step 2: Write down the equation for the magnetic force on a current-carrying conductor

F = BIL sin θ

Step 3: Substitute in values and calculate

F = (80 × 10⁻³) × (0.87) × (1.4) × sin (30) = 0.04872 = 0.049 N (2 s.f)