Calculate the angle of incident of beam radiation on a surface located at 45 degree north and 89.4 degree longitude, at 2:41 pm clock time (local standard time) on February 3, if the surface is tilted 30 degree from the horizontal pointed 10 degree east of south?

To calculate the angle of incident of beam radiation, we need to consider the location, time, and the orientation of the surface. Here's how you can proceed:

Step 1: Determine the solar altitude angle (θs)
To find the solar altitude angle at a given time, you need the latitude of the location and the day of the year. Since you already have the latitude (45 degrees north), we'll focus on the day of the year.

To calculate the day of the year, you can use the formula:
Day of the year = (Number of days in the previous months) + Current day of the month

Since the date is February 3, we only need to consider the days in January. There are 31 days in January, so:
Day of the year = 31 + 3 = 34

Next, we can calculate the solar altitude angle (θs) using the following formula:
θs = 90° - latitude + δ + c

Where δ represents the solar declination angle and c represents the correction for the equation of time.

Step 2: Calculate the solar declination angle (δ)
Solar declination is defined as the angle between the rays of the sun and the plane of the Earth's equator. It changes throughout the year due to the Earth's tilt. The equation for solar declination is:

δ = 23.45° * sin(360° / 365 * (284 + n))

Where n represents the day of the year.

So, we substitute n with the day of the year (34 in this case) and calculate the solar declination angle (δ).

Step 3: Calculate the equation of time correction (c)
The equation of time correction accounts for the difference between solar time and the local standard time. The equation of time depends on the day of the year and can be obtained from various online sources or formula calculations.

Step 4: Calculate the angle of incidence (θi)
Finally, to calculate the angle of incidence (θi), we can use the following formula:

θi = θs - (90° - surface tilt angle) * cos(azimuth)

Where surface tilt angle is 30 degrees (tilted 30 degrees from the horizontal), and azimuth is the direction the surface is pointed, 10 degrees east of south.

By following these steps, you can calculate the angle of incident of beam radiation on the given surface at the specified time and location.