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Two non conducting spherical shellsare uniformly charged .one shell S1 having radius 5 m and charge -2 micro coloumb has a centre at (-1,2,-1)and the other shell S2 has a radius 6 m and charge 3micro coloumb and centre at (-2,1,-1).FindFind the electric field(in vector form) at point (1,-1,3)?
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Two non conducting spherical shellsare uniformly charged .one shell S1...
Problem Statement:
Two non-conducting spherical shells are uniformly charged. One shell S1 having radius 5 m and charge -2 micro coulombs has a centre at (-1, 2, -1) and the other shell S2 has a radius 6 m and charge 3 micro coulombs and centre at (-2, 1, -1). Find the electric field (in vector form) at point (1, -1, 3).
Solution:
Step 1: Calculate the electric field due to each shell separately.
We know that the electric field due to a uniformly charged spherical shell is given by:
E = kQ/r^2
where k is the Coulomb constant, Q is the charge on the shell and r is the distance from the centre of the shell.
Electric field due to S1:
E1 = kQ1/r1^2
= (9x10^9 Nm^2/C^2)(-2x10^-6 C)/(5m)^2
= -72x10^3 N/C
Electric field due to S2:
E2 = kQ2/r2^2
= (9x10^9 Nm^2/C^2)(3x10^-6 C)/(6m)^2
= 22.5x10^3 N/C
Step 2: Calculate the distance and unit vector from the point P(1, -1, 3) to the centres of the shells.
Distance from P to the centre of S1:
r1 = ((-1-1)^2 + (2+1)^2 + (-1-3)^2)^(1/2)
= 4.36 m
Distance from P to the centre of S2:
r2 = ((-2-1)^2 + (1+1)^2 + (-1-3)^2)^(1/2)
= 5.92 m
Unit vector from P to the centre of S1:
v1 = (1, -1, 3) - (-1, 2, -1)
= (2, -3, 4)
v1_hat = v1/|v1|
= (2/7, -3/7, 4/7)
Unit vector from P to the centre of S2:
v2 = (1, -1, 3) - (-2, 1, -1)
= (3, -2, 4)
v2_hat = v2/|v2|
= (3/7, -2/7, 4/7)
Step 3: Calculate the net electric field at point P.
The net electric field at point P due to both shells is given by the vector sum of the electric fields due to each shell:
E_net = E1*v1_hat + E2*v2_hat
= (-72x10^3 N/C)(2/7, -3/7, 4/7) + (22.5x10^3 N/C)(3/7, -2/7, 4/7)
= (-13.7x10^3 N/C, 16.3x10^3
Two non-conducting spherical shells are uniformly charged. One shell S1 having radius 5 m and charge -2 micro coulombs has a centre at (-1, 2, -1) and the other shell S2 has a radius 6 m and charge 3 micro coulombs and centre at (-2, 1, -1). Find the electric field (in vector form) at point (1, -1, 3).
Solution:
Step 1: Calculate the electric field due to each shell separately.
We know that the electric field due to a uniformly charged spherical shell is given by:
E = kQ/r^2
where k is the Coulomb constant, Q is the charge on the shell and r is the distance from the centre of the shell.
Electric field due to S1:
E1 = kQ1/r1^2
= (9x10^9 Nm^2/C^2)(-2x10^-6 C)/(5m)^2
= -72x10^3 N/C
Electric field due to S2:
E2 = kQ2/r2^2
= (9x10^9 Nm^2/C^2)(3x10^-6 C)/(6m)^2
= 22.5x10^3 N/C
Step 2: Calculate the distance and unit vector from the point P(1, -1, 3) to the centres of the shells.
Distance from P to the centre of S1:
r1 = ((-1-1)^2 + (2+1)^2 + (-1-3)^2)^(1/2)
= 4.36 m
Distance from P to the centre of S2:
r2 = ((-2-1)^2 + (1+1)^2 + (-1-3)^2)^(1/2)
= 5.92 m
Unit vector from P to the centre of S1:
v1 = (1, -1, 3) - (-1, 2, -1)
= (2, -3, 4)
v1_hat = v1/|v1|
= (2/7, -3/7, 4/7)
Unit vector from P to the centre of S2:
v2 = (1, -1, 3) - (-2, 1, -1)
= (3, -2, 4)
v2_hat = v2/|v2|
= (3/7, -2/7, 4/7)
Step 3: Calculate the net electric field at point P.
The net electric field at point P due to both shells is given by the vector sum of the electric fields due to each shell:
E_net = E1*v1_hat + E2*v2_hat
= (-72x10^3 N/C)(2/7, -3/7, 4/7) + (22.5x10^3 N/C)(3/7, -2/7, 4/7)
= (-13.7x10^3 N/C, 16.3x10^3
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Two non conducting spherical shellsare uniformly charged .one shell S1 having radius 5 m and charge -2 micro coloumb has a centre at (-1,2,-1)and the other shell S2 has a radius 6 m and charge 3micro coloumb and centre at (-2,1,-1).FindFind the electric field(in vector form) at point (1,-1,3)?
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