The Magnitude and Direction of the Force on a Charge Moving in a Magnetic Field
- When a charged particle moves through a magnetic field , it experiences a force known as the magnetic force.
- This force depends on the charge, velocity, magnetic field strength, and the angle between the velocity and the magnetic field.
The Magnitude of the Magnetic Force
The magnitude of the magnetic force on a moving charge is given by the equation:
$$F = qvB \sin \theta$$
where:
- $F$ is the magnetic force (in newtons, N)
- $q$ is the charge of the particle (in coulombs, C)
- $v$ is the speed of the particle (in meters per second, m/s)
- $B$ is the magnetic field strength (in teslas, T)
- $\theta$ is the angle between the velocity vector and the magnetic field vector
Note
- The force is maximized when the charge moves perpendicular to the magnetic field ($\theta = 90^\circ$), as $\sin 90^\circ = 1$.
- If the charge moves parallel to the field ($\theta = 0^\circ$ or $180^\circ$), the force is zero because $\sin 0^\circ = 0$.
The Direction of the Magnetic Force
The direction of the magnetic force is determined by the right-hand rule:
- Point your fingers in the direction of the velocity ($\vec{v}$).
- Curl your fingers toward the direction of the magnetic field ($\vec{B}$).
- Your thumb points in the direction of the force ($\vec{F}$) for a positive charge .
- For a negative charge , the force is in the opposite direction.
Otherwise, you can also use Fleming's left-hand rule:
- Stretch out your thumb, forefinger, and middle finger so that they are mutually perpendicular.
- Point your forefinger in the direction of the magnetic field ($\vec{B}$).
- Point your middle finger in the direction of the current ($\vec{I}$).
- Your thumb then points in the direction of the force ($\vec{F}$) acting on the conductor.
Example
- A proton ($q = +1.6 \times 10^{-19} \text{ C}$) moves at $2 \times 10^6 \text{ m s}^{-1}$ perpendicular to a magnetic field of $0.5 \text{ T}$.
- The force on the proton is:
$$F = qvB \sin \theta = (1.6 \times 10^{-19} \text{ C})(2 \times 10^6 \text{ m/s})(0.5 \text{ T}) \sin 90^\circ = 1.6 \times 10^{-13} \text{ N}$$
The Magnitude and Direction of the Force on a Current-Carrying Conductor in a Magnetic Field
- A current-carrying conductor in a magnetic field also experiences a force.
- This force is the result of the magnetic forces acting on the moving charges (electrons) within the conductor.
The Magnitude of the Magnetic Force
The magnitude of the force on a straight conductor of length $L$ carrying a current $I$ in a magnetic field $B$ is given by:
$$F = BIL \sin \theta$$
where:
- $F$ is the magnetic force (in newtons, N)
- $B$ is the magnetic field strength (in teslas, T)
- $I$ is the current (in amperes, A)
- $L$ is the length of the conductor in the field (in meters, m)
- $\theta$ is the angle between the current direction and the magnetic field
Note
- The force is maximized when the conductor is perpendicular to the magnetic field ($\theta = 90^\circ$).
- If the conductor is parallel to the field ($\theta = 0^\circ$ or $180^\circ$), the force is zero.
The Direction of the Magnetic Force
The direction of the force on a current-carrying conductor is also determined by the right-hand rule:
- Point your thumb in the direction of the current ($I$).
- Point your fingers in the direction of the magnetic field ($\vec{B}$).
- Your palm faces the direction of the force ($\vec{F}$).
Example
- A wire of length $0.5$ m carries a current of $4$ A perpendicular to a magnetic field of $0.2$ T.
- The force on the wire is:
$$F = BIL \sin \theta = (0.2 \text{ T})(4 \text{ A})(0.5 \text{ m}) \sin 90^\circ = 0.4 \text{ N}$$
The Force per Unit Length Between Parallel Wires
- When two parallel wires carry currents, they exert forces on each other due to their magnetic fields.
- This phenomenon is the basis for the definition of the ampere, the unit of electric current.
The Formula for Force per Unit Length
The force per unit length between two parallel wires separated by a distance $r$ and carrying currents $I_1$ and $I_2$ is given by:
$$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi r}$$
where:
- $F$ is the force between the wires (in newtons, N)
- $L$ is the length of the wires (in meters, m)
- $\mu_0$ is the permeability of free space ($\mu_0 = 4\pi \times 10^{-7} \text{ T m A}^{-1}$)
- $I_1$ and $I_2$ are the currents in the wires (in amperes, A)
- $r$ is the separation between the wires (in meters, m)
The Direction of the Force
- If the currents are in the same direction , the wires attract each other.
- If the currents are in opposite directions , the wires repel each other.
Example
- Two parallel wires $0.1$ m apart carry currents of $3$ A and $5$ A in the same direction.
- The force per unit length between the wires is:
$$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi r} = \frac{(4\pi \times 10^{-7} \text{ T m A}^{-1})(3 \text{ A})(5 \text{ A})}{2\pi (0.1 \text{ m})} = 3 \times 10^{-5} \text{ N m}^{-1}$$
Why Do Moving Charges Experience a Force?
- The magnetic force arises because moving charges create a magnetic field, and this field interacts with external magnetic fields.
- The force is always perpendicular to both the velocity of the charge and the magnetic field, causing the charge to move in a circular or helical path.
Hint
- The magnetic force does no work on the charge because it is always perpendicular to the velocity.
- This means the speed of the charge remains constant, but its direction changes.
Linking Magnetic Force to Current-Carrying Conductors
- The force on a current-carrying conductor can be understood by considering the forces on the individual charges moving within the conductor.
- The total force on the conductor is the sum of the forces on all the moving charges.
Example
- A wire of length $0.2$ m carries a current of $2$ A at an angle of $30^\circ$ to a magnetic field of $0.3$ T.
- The force on the wire is:
$$F = BIL \sin \theta = (0.3 \text{ T})(2 \text{ A})(0.2 \text{ m}) \sin 30^\circ = 0.06 \text{ N}$$
Self review
- An electron ($q = -1.6 \times 10^{-19}$ C) moves at $1 \times 10^7 \text{ m s}^{-1}$ perpendicular to a magnetic field of $0.1$ T. What is the magnitude and direction of the force on the electron?
- A wire of length $0.3$ m carries a current of $5$ A at an angle of $60^\circ$ to a magnetic field of $0.4$ T. What is the force on the wire?
- Two parallel wires $0.05$ m apart carry currents of $4$ A and $6$ A in opposite directions. What is the force per unit length between the wires? Are they attracting or repelling each other?
