Calculating Current Density in an Electron Beam

To calculate the current density in an electron beam, we need to determine the number of electrons passing through a perpendicular cross-section per unit time and the area of that cross-section.

Given:
- Aperture area of the electron beam: 2 mm^2
- Number of electrons passing through any perpendicular cross-section per second: 7×10^6

Step 1: Convert the aperture area to square meters
The given aperture area is in square millimeters. To calculate the current density, we need the area in square meters.
1 square millimeter (mm^2) = 1 × 10^(-6) square meters (m^2)

Converting the aperture area:
2 mm^2 × (1 × 10^(-6) m^2 / 1 mm^2) = 2 × 10^(-6) m^2

Therefore, the aperture area is 2 × 10^(-6) square meters.

Step 2: Calculate the current density
Current density (J) is defined as the amount of current flowing per unit area. It is given by the formula:
J = I / A

where:
J is the current density (in Amperes per square meter, A/m^2)
I is the current (in Amperes, A)
A is the area (in square meters, m^2)

In this case, the current is the number of electrons passing through the cross-section per second. So, we can rewrite the formula as:
J = (number of electrons / time) / A

Plugging in the given values:
J = (7×10^6 electrons/s) / (2 × 10^(-6) m^2)

Simplifying:
J = 7×10^6 / (2 × 10^(-6))
J = 3.5 × 10^12 A/m^2

Therefore, the current density in the electron beam is 3.5 × 10^12 Amperes per square meter.

Summary:
The current density in the electron beam is calculated by dividing the number of electrons passing through a perpendicular cross-section per unit time by the area of that cross-section. In this case, the given aperture area is converted to square meters, and then the current density is calculated using the formula J = (number of electrons / time) / A. The resulting current density is 3.5 × 10^12 Amperes per square meter.

Use formula J(current density) =current (I) /area so current =7*10power 6 *1.6*10 power - 19