{"id":1172,"date":"2019-02-25T23:08:59","date_gmt":"2019-02-26T06:08:59","guid":{"rendered":"https:\/\/sites.evergreen.edu\/psam1819\/?p=1172"},"modified":"2019-02-25T23:10:07","modified_gmt":"2019-02-26T06:10:07","slug":"feb-25-em-lecture-followup","status":"publish","type":"post","link":"https:\/\/sites.evergreen.edu\/psam1819\/feb-25-em-lecture-followup\/","title":{"rendered":"Feb 25 EM Lecture Followup"},"content":{"rendered":"

In my lecture it was not clear enough how to use Equation 4.5 to calculate force. I found what I consider a clear explanation online<\/a> (Equation 5 on this web page and the example immediately following it).<\/p>\n

The way the formula is supposed to work is as follows:<\/p>\n

Each component of E<\/strong> is some scalar function of position r<\/strong>. To find the x<\/em> component of the force on a dipole, start by taking the gradient of the function E_x<\/em>(r<\/strong>).<\/p>\n

Now take the dot product of p<\/strong> (the full p<\/strong> vector!) with the gradient you just found. The result is the x<\/em> component of the force on the dipole.<\/p>\n

Repeat for the y<\/em> and z<\/em> components of E<\/strong> to find the y<\/em> and z<\/em> components of F<\/strong>.<\/p>\n

The example on the we page linked about offers an electric field that is basically\u00a0E<\/strong>=k(y x_hat + x y_hat). To find the x component of force, take the gradient of the function ky, which yields the vector k y_hat. Now dot this with your dipole vector p<\/strong>; this will give k p_y\u00a0 (y_hat dot y_hat) = k p_y.<\/p>\n

Similarly, to find the y component of force, take the gradient of the y component of E<\/strong>. This gives k x_hat. The dot product of this gradient vector with p\u00a0<\/strong>will therefore be kp_x.<\/p>\n

Below is the PowerPoint from this afternoon (with a new Slide 10 on this topic). Note that we skipped Problem 4.7 and Problem 4.14 in class.<\/p>\n