The magnitude of the net electric field at the origin due to the given distribution of charge is E_net = 4kQ/πR².
Given:
A semi-circle distribution with radius 'r'
Charges '+Q' and '-Q'
Calculation of net electric field:
Step 1:
The electric field for a charge is given as:
E = (kλ/R) i + (kλ/R) j - ( 1 )
where, k is Coulomb's force constant
λ is charge density
R is separation distance between the charges
i & j are unit vectors for x-axis and y-axis respectively
Step 2:
The electric field due to charge +Q is given as:
E₁ = (kλ/R) i - (kλ/R) j - ( 2 )
The electric field due to charge -Q is given as:
E₂ = (kλ/R) i + (kλ/R) j - ( 3 )
Step 3:
Thus, the net electric field will be given as:
E_net = E₁ + E₂
Applying values in above equation, we get:
E_net = (kλ/R) i -(kλ/R) j + (kλ/R) i + (kλ/R) j
=( kλ/R) [ i - j + i + j ]
= 2i (kλ/R) -( 4 )
Step 4:
Now, we know that the linear charge density is given as:
λ = Charge/length
= Q/(πR/2)
= 2Q/πR
Applying the value in equation 4, we get:
E_net = 2i (k(2Q/πR)/R)
= 4i (kQ/πR²)
= 4kQ/πR²
Therefore, the net electric field due to the given charge distribution is of magnitude 4kQ/πR² directed along the positive x-axis.
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