Cours et Exercices

∎ 1. The potential field created by a disc at a point on its axis of revolution . ~ ∎ ( The Solution )   ∎ 2. Potential and field created by a charged hemisphere surface ~ ∎ ( the solution ) ∎ 3. The potential created by a portion of a cone ~ ∎ ( The solution ) ∎ 4. Field Lines ►See the list of electromagnetic corrected exercises

∎ Back to exercise   Consider a surface element S on the truncated cone centered about the point P. The Elemental electrostatic potential created at the point O of the axis Ox by a surface element dS centered about the point P is expressed as: so : We get :   By integration: We get : ∎ Back to exercise ∎ Back to the list of electromagnetism corrected exercises

Consider a portion of a cone ,of half-angle with the vertex a and  limit rays R1 and R2 (R1 <R2). This system is loaded on the surface with non-uniform density: a is a constant homogeneous to a length and r the radius of the cone at a point on its axis of symmetry. Determine the electrostatic potential  at the vertex O of the cone. ► See the solution ► Back to the corrected exercises of electromagnetism 

Consider a disk its center O and its radius R. This system is loaded on the surface with the surface density:   a is a constant homogeneous to a length and r is the distance of a point P on the surface of the disc to its center . Determine the electrostatic potential at a point M of the axis  of revolution Oz of this disc. For the calculation is carried out the change of variable r = z u sh. Deduce the expression of the electrostatic field in point M. ► See the solution ► Back to the exercises of electromagnetism list  

∎ Back to exercise 1. The potential at a point M of the axis. Consider a surface area element of the disk centered at a point P. The electrostatic potential created at a point M of the axis Oz is expressed as: Since r and θ variables are separated : For the calculation is performed following variable change: For : On the other hand : We get : Taking the zero potential at infinity is obtained: 2. Field at a point M of the axis. The z axis is axis of symmetry of the charge distribution. The field at a point M of the axis is carried by this axis: We must therefore calculate the derivative of the function . We have: x = y where sh As   ∎ Back to exercise ∎ Back to the list of electromagnetism exercises

► 1. Field the center of a ring having an opening. ~  ► (The solution ) ► 2. Field created by a hemisphere surface   charge . ~   ► ( The Solution )   ► 3. field created by a portion of a cone . ~   ► (The Solution ) ► 4. Field created by a disc at a point on its axis. ~    ► (The Solution )   ► 5. electrostatic field created by an electrified segment. ~   ► (The solution )   ► 6. Electrostatic field created by a hemisphere surface charge. ~   ► ( The solution )   ► 7. Electric field on the axis of a system (-q, + q) ~   ► ( the solution ) ►See the list of electromagnetic corrected exercises

►Back to the Exercise Field created by a portion of a cone. The planes passing through the axis Ox are the charge distribution of planes of symmetry. The electrostatic field must belong simultaneously to all of these plans therefore their intersection: the field has its direction carried by the axis Ox Consider a surface element S on the truncated cone centered about the point P. The figure in the plane of the sheet is as follows: This surface element creates an elementary field O: Only the projection of this vector onto axis Ox contributes to the field at the point O: L is expressed as a function of r: We get : The field O is: ►Back to exercise ►See the list of exercises   

Field created by a portion of a cone. Consider a portion of a cone of half-angle at the apex a and R1 limit rays and R2 (R1 <R2). This system is loaded on the surface with non-uniform density: . a is a constant homogeneous to a length and r the radius of the cone at a point on its axis of symmetry. Determine the electrostatic field at the top of the cone O.   ► see solution ►See the list of exercises  

Field generated by a hemisphere surface charge. Considering a half sphere of center O, of radius R, uniformly charged surface with the surface density s. 1. Determine the electrostatic field at the point O. 2. In the same issue in a point M of the axis Oz of symmetry of this hemisphere. Find the result of Question 1. ► see solution ►See All the list of the Exercises  

►Back to exercise 1. O electrostatic field. The z axis is the axis of symmetry of load distribution. Field O is carried by this axis. Consider a surface element of the half sphere centered at a point P. Elemental electrostatic field created by this element to the point O of the axis Oz is expressed as: Only the projection of this vector onto axis Oz contributes to the field at the point O: The field component O is: 2. Electrostatic Field in M. The z axis is the axis of symmetry of load distribution. The field M is carried by this axis. Consider a surface element of the half sphere centered at a point P. Elemental electrostatic field created by this element at the point M of the axis Oz is expressed as: Only the projection of this vector onto axis Oz contributes to the field at point M: Gold : and where: By integration: Gold : Either: ...