explain biot-savart,law.an electron with velocity 3*107ms1 describe a circular path in a uniform magnetic field of 0.2T,perpendicular to it.calculate the radius of path
The Biot-Savart law is a fundamental principle in electromagnetism that relates the magnetic field created by a current-carrying wire to the magnitude and direction of the current. This law helps us understand how a magnetic field is produced due to a moving charge.
To explain the Biot-Savart law, let's consider a current-carrying wire. The law states that the magnetic field at a certain point due to a short segment of the wire can be determined by the following equation:
dB = (μ₀ / 4π) * (I * dl x r) / r²
In this equation:
- dB represents the magnetic field element at a specific point
- μ₀ is the permeability of free space, approximately 4π x 10⁻⁷ T·m/A
- I is the current flowing through the wire
- dl is the small length element of the wire
- r is the displacement vector from the wire element to the point
Now, let's calculate the radius of a circular path of an electron moving in a uniform magnetic field using the given information.
Given:
- Velocity of the electron (v) = 3 x 10⁷ m/s
- Magnetic field (B) = 0.2 T
When a charged particle moves perpendicular to a magnetic field, it experiences a force called the magnetic Lorentz force, given by the equation:
F = q * (v x B)
In this equation:
- F is the magnetic force on the particle
- q is the charge of the particle
- v is the velocity vector of the particle
- B is the magnetic field vector
Since the electron has a negative charge, the force will act in the opposite direction to the velocity, deflecting the electron into a circular path. The centripetal force acting on the electron is provided by this magnetic force:
F = mv² / r
In this equation:
- m is the mass of the electron
- v is the velocity of the electron
- r is the radius of the circular path
Equating the two equations for force, we can solve for the radius:
mv² / r = q * (v x B)
Using the magnitude of the charge of an electron (q = 1.6 x 10⁻¹⁹ C) and the mass of an electron (m = 9.1 x 10⁻³¹ kg), we can substitute these values into the equation:
(9.1 x 10⁻³¹ kg) * (3 x 10⁷ m/s)² / r = (1.6 x 10⁻¹⁹ C) * (3 x 10⁷ m/s) * (0.2 T)
Simplifying this equation, we can find the value of r.