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A point charge q is placed at origin. Calculate the electric flux through the open hemispherical surface : (x-a) 2 + y 2 +z 2 = a 2 , x ≥ a.

A point charge q is placed at origin. Calculate the electric flux through the open hemispherical surface : (x-a) 2 + y2 +z 2 = a2 ,  x ≥ a.

Grade:12

1 Answers

Pratham Ashish
17 Points
12 years ago

Ø4589-1424_4788_8.bmp

electri flux trough the hemisphere will be same as through the circular surface at its mouth , coz they shere same field lines

flux through the circular surface ,

 consider a thin circular strip of width dr at a distance r from the centre

dØ = E. dA

     = q /4¶ε (a^2 + r ^2 )  . 2¶r dr  cosθ

     = -q r ( a/ √ a^2 + r^2 ) / 2  ε (a^2 + r ^2 )              { cosθ  = - a/ √ a^2 + r^2}

     =- (q /2ε ) ar / ( a^2 + r^2 ) ^3/2  dr

Φ   = -∫ (q /2ε ) ar / ( a^2 + r^2 ) ^3/2  dr

       = (q /2ε ) a / √ a^2 + r^2  ]

       =  - (q /2ε )  (  1 - 1/√2  ) 

flux through hemisphere  = - Φ

                                     =   (q /2ε )  (  1 - 1/√2  )    

 

 

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