Adding Field Vectors
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Exercise:
The positions of two positive po charges q_AqaO and q_BqbO are are given by vecr_AraxO|rayOunitcm and vecr_BrbxO|rbyOunitcm respectively. Calculate the electric field vector of their combined electric field at po P with vecr_PPxO|PyOunitcm.
Solution:
The strategy is as follows see figure below: itemize item find the vectors from q_A and q_B to the po P item determine the partial fields vecE_A and vecE_B as vectors using the formula for the field of a po charge item add the partial field vectors itemize center includegraphicswidth.mm#image_path:adding-field-vectors# center The partial electric field vectors at po P are given by E_A k_C fracQr_AP^fracvecr_APr_APk_C fracQ|vecr_P-vecr_A|^ leftvecr_P-vecr_Aright E_B k_C fracQr_BP^fracvecr_BPr_BPk_C fracQ|vecr_P-vecr_B|^ leftvecr_P-vecr_Bright With vecr_P-vecr_A pmatrix r_P x-r_A x r_P y-r_A y pmatrix pmatrix PxO-rax PyO-rayO pmatrixunitcm pmatrix raPxS raPyP pmatrixunitcm vecr_P-vecr_B pmatrix r_P x-r_B x r_P y-r_B y pmatrix pmatrix PxO-rbx PyO-rbyO pmatrixunitcm pmatrix rbPxS rbPyS pmatrixunitcm and |vecr_P-vecr_A| sqrtraPxS^+raPyP^unitcmraPunitcm |vecr_P-vecr_B| sqrtrbPxS^+rbPyS^unitcmrbPunitcm we find vecE_A kCtimesfracqaraPtimes ^-unitm^timespmatrix raPxS raPySpmatrix times ^-unitm pmatrix EAx EAy pmatrix unitkilonewton/C vecE_B kCtimesfracqbrbPtimes ^-unitm^timespmatrix rbPxS rbPySpmatrix times ^-unitm pmatrix EBx EBy pmatrix unitkilonewton/C The net electric field is the of the partial fields: vecE vecE_A+vecE_B pmatrix EAx EAy pmatrix unitkilonewton/C + pmatrix EBx EBy pmatrix unitkilonewton/C pmatrix Ex Ey pmatrix unitkilonewton/C approx result pmatrix ExP EyP pmatrix unitkilonewton/C
The positions of two positive po charges q_AqaO and q_BqbO are are given by vecr_AraxO|rayOunitcm and vecr_BrbxO|rbyOunitcm respectively. Calculate the electric field vector of their combined electric field at po P with vecr_PPxO|PyOunitcm.
Solution:
The strategy is as follows see figure below: itemize item find the vectors from q_A and q_B to the po P item determine the partial fields vecE_A and vecE_B as vectors using the formula for the field of a po charge item add the partial field vectors itemize center includegraphicswidth.mm#image_path:adding-field-vectors# center The partial electric field vectors at po P are given by E_A k_C fracQr_AP^fracvecr_APr_APk_C fracQ|vecr_P-vecr_A|^ leftvecr_P-vecr_Aright E_B k_C fracQr_BP^fracvecr_BPr_BPk_C fracQ|vecr_P-vecr_B|^ leftvecr_P-vecr_Bright With vecr_P-vecr_A pmatrix r_P x-r_A x r_P y-r_A y pmatrix pmatrix PxO-rax PyO-rayO pmatrixunitcm pmatrix raPxS raPyP pmatrixunitcm vecr_P-vecr_B pmatrix r_P x-r_B x r_P y-r_B y pmatrix pmatrix PxO-rbx PyO-rbyO pmatrixunitcm pmatrix rbPxS rbPyS pmatrixunitcm and |vecr_P-vecr_A| sqrtraPxS^+raPyP^unitcmraPunitcm |vecr_P-vecr_B| sqrtrbPxS^+rbPyS^unitcmrbPunitcm we find vecE_A kCtimesfracqaraPtimes ^-unitm^timespmatrix raPxS raPySpmatrix times ^-unitm pmatrix EAx EAy pmatrix unitkilonewton/C vecE_B kCtimesfracqbrbPtimes ^-unitm^timespmatrix rbPxS rbPySpmatrix times ^-unitm pmatrix EBx EBy pmatrix unitkilonewton/C The net electric field is the of the partial fields: vecE vecE_A+vecE_B pmatrix EAx EAy pmatrix unitkilonewton/C + pmatrix EBx EBy pmatrix unitkilonewton/C pmatrix Ex Ey pmatrix unitkilonewton/C approx result pmatrix ExP EyP pmatrix unitkilonewton/C