91Ó°ÊÓ

The space around the charge in which another charge experiences some electrostatic force is known as electric field. It is a vector quantity. The expression for the electric field is,

\(E = \frac{{Kq}}{{{r^2}}}\)

Here, \(K\) is the electrostatic force constant, \(q\) is the charge and \(r\) is the distance of the point of consideration from the charge \(q\).

If a number of electric field acts a given point, the net electric field will be equal to the vector sum of individual field.

Due to symmetry the electric field at the center of the square will be straight up, since \({q_a}\) and \({q_b}\) are negative and \({q_c}\) and \({q_d}\) are positive with same magnitude. The resultant of the electric field due to \({q_a}\) and \({q_b}\) will be straight up, and the resultant of the electric field due to \({q_c}\) and \({q_d}\) will be straight up.

Hence, the direction of electric field at the center of the square is straight up.

The electric field at the center of the square is represented as,

Electric field at the center of the square

Here, \({E_a}\) is the electric field at the center of the square due to charge \({q_a}\), \({E_b}\) is the electric field at the center of the square due to charge \({q_b}\), \({E_c}\) is the electric field at the center of the square due to charge \({q_c}\), and \({E_d}\) is the electric field at the center of the square due to charge \({q_d}\).

The distance of test charge \(q\) from the charges \({q_a}\), \({q_b}\), \({q_c}\) and \({q_d}\) is,

\(r = \frac{a}{{\sqrt 2 }}\)

Here, \(a\) is the side of the square \(\left( {a = {\rm{5}}{\rm{.00 cm}}} \right)\).

Substituting all known values,

\(\begin{array}{c}r = \frac{{5.00{\rm{ cm}}}}{{\sqrt 2 }}\\ \approx 3.54{\rm{ cm}}\end{array}\)

The magnitude of the electric field at the center of the square due to charge \({q_a}\),

\({E_a} = \frac{{K{q_a}}}{{{r^2}}}\)

Here, \(K\) is the electrostatic force constant \(\left( {K = {\rm{9 \times 1}}{{\rm{0}}^{\rm{9}}}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right)\), \({q_a}\) is the magnitude of the charge \(\left( {{q_a} = {\rm{1}}{\rm{.00 }}\mu {\rm{C}}} \right)\), and \(r\) is the distance of charge \({q_a}\) from the center of the square \(\left( {r = 3.54{\rm{ cm}}} \right)\).

Substituting all known values,

\(\begin{array}{c}{E_a} = \frac{{\left( {{\rm{9}} \times {\rm{1}}{{\rm{0}}^{\rm{9}}}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right) \times \left( {{\rm{1}}{\rm{.00 }}\mu {\rm{C}}} \right)}}{{{{\left( {{\rm{3}}{\rm{.54 cm}}} \right)}^{\rm{2}}}}}\\ = \frac{{\left( {{\rm{9}} \times {\rm{1}}{{\rm{0}}^{\rm{9}}}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right) \times \left( {{\rm{1}}{\rm{.00 \mu C}}} \right) \times \left( {\frac{{{\rm{1}}{{\rm{0}}^{{\rm{ - 6}}}}{\rm{ C}}}}{{{\rm{1 \mu C}}}}} \right)}}{{{{\left[ {\left( {{\rm{3}}{\rm{.54 cm}}} \right) \times \left( {\frac{{{\rm{1}}{{\rm{0}}^{{\rm{ - 2}}}}{\rm{ m}}}}{{{\rm{1 cm}}}}} \right)} \right]}^{\rm{2}}}}}\\ = {\rm{7}}{\rm{.18}} \times {\rm{1}}{{\rm{0}}^{\rm{6}}}{\rm{ N/C}}\end{array}\)

The magnitude of the electric field at the center of the square due to charge \({q_b}\),

\({E_b} = \frac{{K{q_b}}}{{{r^2}}}\)

Here, \(K\) is the electrostatic force constant \(\left( {K = {\rm{9 \times 1}}{{\rm{0}}^{\rm{9}}}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right)\), \({q_b}\) is the magnitude of the charge \(\left( {{q_b} = 1.00{\rm{ }}\mu {\rm{C}}} \right)\), and \(r\) is the distance of charge \({q_b}\) from the center of the square \(\left( {r = 3.54{\rm{ cm}}} \right)\).

Substituting all known values,

\(\begin{array}{c}{E_b} = \frac{{\left( {{\rm{9}} \times {\rm{1}}{{\rm{0}}^{\rm{9}}}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right) \times \left( {{\rm{1}}{\rm{.00 }}\mu {\rm{C}}} \right)}}{{{{\left( {{\rm{3}}{\rm{.54 cm}}} \right)}^{\rm{2}}}}}\\ = \frac{{\left( {{\rm{9}} \times {\rm{1}}{{\rm{0}}^{\rm{9}}}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right) \times \left( {{\rm{1}}{\rm{.00 \mu C}}} \right) \times \left( {\frac{{{\rm{1}}{{\rm{0}}^{{\rm{ - 6}}}}{\rm{ C}}}}{{{\rm{1 \mu C}}}}} \right)}}{{{{\left[ {\left( {{\rm{3}}{\rm{.54 cm}}} \right) \times \left( {\frac{{{\rm{1}}{{\rm{0}}^{{\rm{ - 2}}}}{\rm{ m}}}}{{{\rm{1 cm}}}}} \right)} \right]}^{\rm{2}}}}}\\ = {\rm{7}}{\rm{.18}} \times {\rm{1}}{{\rm{0}}^{\rm{6}}}{\rm{ N/C}}\end{array}\)

The magnitude of the electric field at the center of the square due to charge \({q_c}\),

\({E_c} = \frac{{K{q_c}}}{{{r^2}}}\)

Here, \(K\) is the electrostatic force constant \(\left( {K = {\rm{9 \times 1}}{{\rm{0}}^{\rm{9}}}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right)\), \({q_c}\) is the magnitude of the charge \(\left( {{q_c} = 1.00{\rm{ }}\mu {\rm{C}}} \right)\), and \(r\) is the distance of charge \({q_c}\) from the center of the square \(\left( {r = 3.54{\rm{ cm}}} \right)\).

Substituting all known values,

\(\begin{array}{c}{E_c} = \frac{{\left( {{\rm{9}} \times {\rm{1}}{{\rm{0}}^{\rm{9}}}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right) \times \left( {{\rm{1}}{\rm{.00 }}\mu {\rm{C}}} \right)}}{{{{\left( {{\rm{3}}{\rm{.54 cm}}} \right)}^{\rm{2}}}}}\\ = \frac{{\left( {{\rm{9}} \times {\rm{1}}{{\rm{0}}^{\rm{9}}}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right) \times \left( {{\rm{1}}{\rm{.00 \mu C}}} \right) \times \left( {\frac{{{\rm{1}}{{\rm{0}}^{{\rm{ - 6}}}}{\rm{ C}}}}{{{\rm{1 \mu C}}}}} \right)}}{{{{\left[ {\left( {{\rm{3}}{\rm{.54 cm}}} \right) \times \left( {\frac{{{\rm{1}}{{\rm{0}}^{{\rm{ - 2}}}}{\rm{ m}}}}{{{\rm{1 cm}}}}} \right)} \right]}^{\rm{2}}}}}\\ = {\rm{7}}{\rm{.18}} \times {\rm{1}}{{\rm{0}}^{\rm{6}}}{\rm{ N/C}}\end{array}\)

The magnitude of the electric field at the center of the square due to charge \({q_d}\),

\({E_d} = \frac{{K{q_d}}}{{{r^2}}}\)

Here, \(K\) is the electrostatic force constant \(\left( {K = {\rm{9 \times 1}}{{\rm{0}}^{\rm{9}}}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right)\), \({q_d}\) is the magnitude of the charge \(\left( {{q_d} = 1.00{\rm{ }}\mu {\rm{C}}} \right)\), and \(r\) is the distance of charge \({q_d}\) from the center of the square \(\left( {r = 3.54{\rm{ cm}}} \right)\).

Substituting all known values,

\(\begin{array}{c}{E_d} = \frac{{\left( {{\rm{9}} \times {\rm{1}}{{\rm{0}}^{\rm{9}}}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right) \times \left( {{\rm{1}}{\rm{.00 }}\mu {\rm{C}}} \right)}}{{{{\left( {{\rm{3}}{\rm{.54 cm}}} \right)}^{\rm{2}}}}}\\ = \frac{{\left( {{\rm{9}} \times {\rm{1}}{{\rm{0}}^{\rm{9}}}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right) \times \left( {{\rm{1}}{\rm{.00 \mu C}}} \right) \times \left( {\frac{{{\rm{1}}{{\rm{0}}^{{\rm{ - 6}}}}{\rm{ C}}}}{{{\rm{1 \mu C}}}}} \right)}}{{{{\left[ {\left( {{\rm{3}}{\rm{.54 cm}}} \right) \times \left( {\frac{{{\rm{1}}{{\rm{0}}^{{\rm{ - 2}}}}{\rm{ m}}}}{{{\rm{1 cm}}}}} \right)} \right]}^{\rm{2}}}}}\\ = {\rm{7}}{\rm{.18}} \times {\rm{1}}{{\rm{0}}^{\rm{6}}}{\rm{ N/C}}\end{array}\)

The field in the horizontal direction is,

\(\begin{array}{c}{E_x} = - {E_a}\sin \left( {45^\circ } \right) + {E_b}\sin \left( {45^\circ } \right) + {E_c}\sin \left( {45^\circ } \right) - {E_d}\sin \left( {45^\circ } \right)\\ = \left( { - {E_a} + {E_b} + {E_c} - {E_d}} \right) \times \sin \left( {45^\circ } \right)\end{array}\)

Substituting all known values,

\(\begin{array}{c}{E_x} = \left[ \begin{array}{l} - \left( {{\rm{7}}{\rm{.18}} \times {\rm{1}}{{\rm{0}}^{\rm{6}}}{\rm{ N/C}}} \right) + \left( {{\rm{7}}{\rm{.18}} \times {\rm{1}}{{\rm{0}}^{\rm{6}}}{\rm{ N/C}}} \right)\\ + \left( {{\rm{7}}{\rm{.18}} \times {\rm{1}}{{\rm{0}}^{\rm{6}}}{\rm{ N/C}}} \right) - \left( {{\rm{7}}{\rm{.18}} \times {\rm{1}}{{\rm{0}}^{\rm{6}}}{\rm{ N/C}}} \right)\end{array} \right] \times {\rm{sin}}\left( {{\rm{45^\circ }}} \right)\\ = 0\end{array}\)

The field in the vertical direction is,

\(\begin{array}{c}{E_y} = {E_a}\cos \left( {45^\circ } \right) + {E_b}\cos \left( {45^\circ } \right) + {E_c}\cos \left( {45^\circ } \right) + {E_d}\cos \left( {45^\circ } \right)\\ = \left( {{E_a} + {E_b} + {E_c} + {E_d}} \right) \times \cos \left( {45^\circ } \right)\end{array}\)

Substituting all known values,

\(\begin{array}{c}{E_y} = \left[ \begin{array}{l}\left( {{\rm{7}}{\rm{.18}} \times {\rm{1}}{{\rm{0}}^{\rm{6}}}{\rm{ N/C}}} \right) + \left( {{\rm{7}}{\rm{.18}} \times {\rm{1}}{{\rm{0}}^{\rm{6}}}{\rm{ N/C}}} \right) + \\\left( {{\rm{7}}{\rm{.18}} \times {\rm{1}}{{\rm{0}}^{\rm{6}}}{\rm{ N}}/{\rm{C}}} \right) + \left( {{\rm{7}}{\rm{.18}} \times {\rm{1}}{{\rm{0}}^{\rm{6}}}{\rm{ N/C}}} \right)\end{array} \right] \times {\rm{cos}}\left( {{\rm{45^\circ }}} \right)\\ = {\rm{2}}{\rm{.03}} \times {\rm{1}}{{\rm{0}}^{\rm{7}}}{\rm{ N}}/{\rm{C}}\end{array}\)

The magnitude of net electric field at the center of the square is,

\(E = \sqrt {E_x^2 + E_y^2} \)

Substituting all known values,

\(\begin{array}{c}E = \sqrt {{{\left( 0 \right)}^2} + {{\left( {2.03 \times {{10}^7}{\rm{ N}}/{\rm{C}}} \right)}^2}} \\ = 2.03 \times {10^7}{\rm{ N}}/{\rm{C}}\end{array}\)

Hence, the magnitude of electric field at the center of the square is \({\rm{2}}{\rm{.03}} \times {\rm{1}}{{\rm{0}}^{\rm{7}}}{\rm{ N}}/{\rm{C}}\).