Electrical Field of a thin Disk © Frits F.M. de Mul E-field of a thin disk

E-field of a thin disk Available : A thin circular disk with radius R and charge density s [C/m2] Question : Calculate E-field in arbitrary points a both sides of the disk E-field of a thin disk

E-field of a thin disk Analysis and symmetry Approach to solution Calculations Conclusions Appendix: angular integration E-field of a thin disk

Analysis and Symmetry 1. Charge distribution: s [C/m2] 2. Coordinate axes: Z-axis = symm. axis, perpend. to disk Z X Y ej j er r ez z 3. Symmetry: cylinder 4. Cylinder coordinates: r, z, j E-field of a thin disk

Analysis, field build-up R Ei Qi ri 3. all Qi’s at ri and ji contribute Ei to E in P 1. XYZ-axes Z Y X 2. Point P on Y-axis P 4. Ei,xy , Ei,z Ei,z Ei,xy 5. expect: S Ei,xy = 0, to be checked !! 6. E = Ez ez only ! E-field of a thin disk

Approach to solution R Z 1. Rings and segments 2. Distributed charges dQ 3. dE er r P 4. dQ = s.dA= s (da.)(a dj) a dj j da 5. z- component only ! dExy dEz 6. E-field of a thin disk

Calculations (1) R Z dQ dE er r P a dj j da dEz zP 1. 2. dQ = s.dA= s (da.)(a dj) 3. 4. E-field of a thin disk

Calculations (2) R Z dQ dE er r P a dj j da dEz zP 4. 6. If R -> infinity : E-field of a thin disk

Conclusions for infinite disk: Z P EP for infinite disk: field strength independent of distance to disk => homogeneous field E-field of a thin disk

Appendix: angular integration (1) Z dQ dE er r P a dj j da dEz zP 1. 2. dQ = s.dA= s (da.)(a dj) 3. E-field of a thin disk

Appendix: angular integration (2) Z dQ dE er r P a dj j da dEz zP 3. the end E-field of a thin disk