4 charges each equal to q are placed at the four corners of a square and a charge q is placed at the center of the square. If the system is in equilibrium then the value of q is
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The given system consist 4 point charges +q placed at 4 corners A, B, C and D of a square of side a and a charge Q at the centre of the square. Any system of charges is in equilibrium if its total electric potential energy is equal to zero.
Hence, Usystem=PE due to all those pairs in which charges placed at the corners of the sidesof square+PE due to all those pairs in which charges are placed at the ends of diagonal of square+PE due to pairs formed by any charge at the corner and charge at the centre.
Usystem=4×14πε0×q2a+2×14πε0×q2a2√+4××14πε0×qQa2√/Usystem=14πε0×4qa[q+q22√+Q2‾√]
The system will be in equilibrium if
Usystem=0
4πε0×4qa[q+q22√+Q2‾√]=0
q+q22√+Q2‾√=0
Q2‾√=−(q+q22√)
Q=−(q2√+q4)
Q=−q(22√+14)
Q=−q4(22‾√+1)
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Hence, Usystem=PE due to all those pairs in which charges placed at the corners of the sidesof square+PE due to all those pairs in which charges are placed at the ends of diagonal of square+PE due to pairs formed by any charge at the corner and charge at the centre.
Usystem=4×14πε0×q2a+2×14πε0×q2a2√+4××14πε0×qQa2√/Usystem=14πε0×4qa[q+q22√+Q2‾√]
The system will be in equilibrium if
Usystem=0
4πε0×4qa[q+q22√+Q2‾√]=0
q+q22√+Q2‾√=0
Q2‾√=−(q+q22√)
Q=−(q2√+q4)
Q=−q(22√+14)
Q=−q4(22‾√+1)
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