An electron travels at a speed of 5.56×107 m/s along the z-axis. What is the magnetic field at the point (0.00 cm,7.00 cm,7.00 cm) when the electron passes through the origin?

The magnetic field B created by a moving charge can be calculated using the Biot-Savart law. The Biot-Savart law states that the magnetic field d𝐵 at a given point in space due to an infinitesimal segment of current is given by:

d𝐵 = μ0/4π * (Id𝐥 × 𝐫/𝐫³)

where:

• μ0 is the permeability of free space (4π × 10^-7 T m/A),
• I is the current,
• d𝐥 is the infinitesimal segment of the current,
• 𝐫 is the position vector from the current element to the point in space,
• 𝐫 is the magnitude of 𝐫.

However, in this case, we have a moving electron, not a current. The relationship between current and charge is given by I = dq/dt, where dq is the charge and dt is the time. For an electron, dq is just the charge of the electron, e = 1.6 × 10^-19 C.

The speed of the electron gives us a way to relate dt and d𝐥. We know that speed v = d𝐥/dt, so dt = d𝐥/v. Substituting this into the expression for current gives I = dq/dt = e*v/d𝐥.

Substituting this into the Biot-Savart law gives:

d𝐵 = μ0/4π * (ev/d𝐥 * d𝐥 × 𝐫/𝐫³) = μ0/4π * (ev × 𝐫/𝐫³)

The direction of d𝐵 is given by the right-hand rule. The electron is moving along the z-axis, so d𝐵 is in the xy-plane. The magnitude of d𝐵 is given by:

|d𝐵| = μ0/4π * e*v/𝐫²

The position vector 𝐫 is (0.00 cm,7.00 cm,7.00 cm) = (0.00 m,0.07 m,0.07 m), so 𝐫 = sqrt((0.00 m)² + (0.07 m)² + (0.07 m)²) = 0.099 m.

Substituting the given values gives:

|d𝐵| = (4π × 10^-7 T m/A) / 4π * (1.6 × 10^-19 C) * (5.56 × 10^7 m/s) / (0.099 m)² = 3.57 × 10^-9 T.

So the magnetic field at the point (0.00 cm,7.00 cm,7.00 cm) when the electron passes through the origin is 3.57 × 10^-9 T.