Charged Wire
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Exercise:
A test charge qtO is placed at rO from a wire with length lwO and a charge of QwO. Calculate the force acting between test charge and wire.
Solution:
Field produced by a wire with length L and charge Q_w i.e. linear charge density lambdaQ_w/L: Er fraclambdapivarepsilon_ r fracQ_wpivarepsilon_ r L Force on the test charge q at distance r from the wire: sscFE q Er FF labeleq:wiron-q fracQwtimesqtpitimesncepstimesrtimeslw F approx resultFP- labelres:wiron-q Remark: This solution uses the approximation for an infinitely long wire rll L. We can also determine the force by calculating the force acting on the wire in the field of the po charge. Since the electric field varies along the wire we have to egrate the partial forces along the wire. The idea is illustrated in the figure below. center includegraphicswidthtextwidth#image_path:forcon-wirin-field-of-po-charg# center For a uniformly charged wire with total charge Q_w and length L the charge on a line segment with length textdy is textdQ Q_w fractextdyL lambda textdy The partial force on the segment is thus dF k_C fracq textdQd^ k_C fracq lambda textdyr^+y^ Because of the symmetry of the situation the force components in the y direction cancel there is a po symmetrical to the one in the figure in the lower half of the wire so we only have to consider the component in the x direction which can be found using similar triangles: fractextdF_xtextdF fracrd Longrightarrow textdF_x textdF fracrd k_C fracq lambda textdyr^+y^ fracrleftr^+y^right^/ k_C q lambda r fractextdyleftr^+y^right^/ For the total force acting on the wire we have to egrate the partial forces: F _-L/^+L/ textdF_x k_C q lambda r _-L/^+L/ fractextdyleftr^+y^right^/ The antiderivative of gy fracleftr^+y^right^/ is as can be verified by showing that G'y gy Gy fracyr^ sqrty^+r^ It follows for the force F k_C q lambda r left fracyr^ sqrty^+r^ right_-L/^+L/ k_C q fracQ_wL r fracLr^ sqrtL^/+r^ FbF labeleq:exact fracpitimesnceps times fracQwtimesqtrtimes sqrtlw^/+r^ Fb approx resultFbP- labelres:exact The numerical result refres:exact agrees nicely with the approximated result in refres:wiron-q since in this case the distance is considerably smaller than the length of the wire. Formally we can approximate expression refeq:exact for r ll L by neglecting the second term in the square root: F &approx fracpivarepsilon_ fracq Q_wr sqrtL^/ fracpivarepsilon_ frac q Q_wr L FbappF which is the same as refeq:wiron-q.
A test charge qtO is placed at rO from a wire with length lwO and a charge of QwO. Calculate the force acting between test charge and wire.
Solution:
Field produced by a wire with length L and charge Q_w i.e. linear charge density lambdaQ_w/L: Er fraclambdapivarepsilon_ r fracQ_wpivarepsilon_ r L Force on the test charge q at distance r from the wire: sscFE q Er FF labeleq:wiron-q fracQwtimesqtpitimesncepstimesrtimeslw F approx resultFP- labelres:wiron-q Remark: This solution uses the approximation for an infinitely long wire rll L. We can also determine the force by calculating the force acting on the wire in the field of the po charge. Since the electric field varies along the wire we have to egrate the partial forces along the wire. The idea is illustrated in the figure below. center includegraphicswidthtextwidth#image_path:forcon-wirin-field-of-po-charg# center For a uniformly charged wire with total charge Q_w and length L the charge on a line segment with length textdy is textdQ Q_w fractextdyL lambda textdy The partial force on the segment is thus dF k_C fracq textdQd^ k_C fracq lambda textdyr^+y^ Because of the symmetry of the situation the force components in the y direction cancel there is a po symmetrical to the one in the figure in the lower half of the wire so we only have to consider the component in the x direction which can be found using similar triangles: fractextdF_xtextdF fracrd Longrightarrow textdF_x textdF fracrd k_C fracq lambda textdyr^+y^ fracrleftr^+y^right^/ k_C q lambda r fractextdyleftr^+y^right^/ For the total force acting on the wire we have to egrate the partial forces: F _-L/^+L/ textdF_x k_C q lambda r _-L/^+L/ fractextdyleftr^+y^right^/ The antiderivative of gy fracleftr^+y^right^/ is as can be verified by showing that G'y gy Gy fracyr^ sqrty^+r^ It follows for the force F k_C q lambda r left fracyr^ sqrty^+r^ right_-L/^+L/ k_C q fracQ_wL r fracLr^ sqrtL^/+r^ FbF labeleq:exact fracpitimesnceps times fracQwtimesqtrtimes sqrtlw^/+r^ Fb approx resultFbP- labelres:exact The numerical result refres:exact agrees nicely with the approximated result in refres:wiron-q since in this case the distance is considerably smaller than the length of the wire. Formally we can approximate expression refeq:exact for r ll L by neglecting the second term in the square root: F &approx fracpivarepsilon_ fracq Q_wr sqrtL^/ fracpivarepsilon_ frac q Q_wr L FbappF which is the same as refeq:wiron-q.