Flux as a Function of Surface Angle
In this activity we find the flux of a surface as a function of its surface angle.
In the picture above we have a bed of nails, which represents an electric field and a metal square which we use to represent a surface. The number of nails (electric field lines) that would go through the surface is varied by adjusting the angle of the angle the surface makes from perpendicular to the electric field. We used a measured height to calculate the angle rather than using a protractor directly and using a calculated column on LoggerPro software to automatically calculate the corresponding angles with respect to the measured heights (h). The formula we used to calculate the angle in degrees (Ref Degrees) was (arcsin(h/hypotenuse))*(360/2pi). The hypotenuse was measured as 6.75 cm. The program automatically calculated for radians therefore we multiplied by the formula to get it in degrees. To get the angle perpendicular to the surface, we needed to adjust the degrees where the flux is negative. This was done using the following for when flux is negative: 90+(90-Ref Degrees). This adjustment help form the graph. We then went ahead and created another calculated column for this angle in radians. The formula was just degrees*(2pi/360). The number of field lines (flux) was plotted with their corresponding angle to observe the relationship between the two.
After plotting the points we added a trendline to the graph. The one that fit it most properly was a sine function (though in this case it has a horizontal shift factor that makes it a cosine). We can see from the graph that when the angle is 0, the flux is also at a maximum (49). When the angle is zero, the surface is perpendicular to the electric field and therefore the flux is at maximum, but when the angle is pi/2 then the flux is zero because the surface is parallel to the electric field. This same behavior is consistent with the cosine function: cos(0)=1, cos(pi/2)=0. The electric field multiplied by the cosine of the angle between it and the surface gives us the flux. Moreover, the cosine also defines a dot product between E vector and dA vector.
Electric Flux Activity
Throughout the simulations, we made some observations about flux as seen on these whiteboard pictures.
