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In physics, the term magnitude can be simply defined as ‘distance’. It indicates the dimension or direction of an object that’s absolute, or relative to the sense that an object is moving in the sense of motion. It’s used to define the dimensions or size of an item.
This problem can be solved using the following concepts: electric force between two points charges, resolution in to rectangular components and net force on an object because of multiple forces acting simultaneously.
In the beginning, you will call four identical point charges A, B,C, and D. The four charges should be placed at the corners of a square with side length L in a clockwise order. Later, convert the vector form of charges A, B, C and D into the electric force on charge B. These three forces on charge A can be added later in vector form. Due to the three additional charges, this force vector is known as net force vector on charge B. Calculate the force vector’s magnitude later.
The electric force F for a charge [katex]{q_1}[/katex].
[katex]F = \frac{{k{q_1}{q_2}}}{{{r^2}}}[/katex]
k refers to the constant of Coulomb and r the distance between them.
The net electric force [katex]{\vec F_{{\rm{net,B}}}}[/katex] is derived from the superposition principle.
[katex]{\vec F_{{\rm{net,B}}}} = {\vec F_{\rm{A}}} + {\vec F_{\rm{C}}} + {\vec F_{\rm{D}}}[/katex]
According to the Pythagoras Theorem, the squared hypotenuse of a right-angle triangle is equal to the sum squares of each side.
Magnitude of [katex]\vec s = {s_x}\hat i + {s_y}\hat j[/katex] two-dimensional vector
[katex]\left| {\vec s} \right| = \sqrt {s_x^2 + s_y^2} [/katex]
Below is the arrangement of four charged at the corners of a square, and the direction of the electric forces acting upon a charge due the other three charges.
Substitute Q to [katex]{q_1}[/katex]
[katex]\begin{array}{c}\\{F_{\rm{A}}} = \frac{{k\left( Q \right)\left( Q \right)}}{{{L^2}}}\\\\ = \frac{{k{Q^2}}}{{{L^2}}}\\\end{array}[/katex]
The vector form for the force on charge B due charge A is
[katex]F = \frac{{k{q_1}{q_2}}}{{{r^2}}}[/katex]
Substitute Q to [katex]{q_1}[/katex]
[katex]F = \frac{{k{q_1}{q_2}}}{{{r^2}}}[/katex]
The vector form for the force on charge A due to charge C, is
[katex]F = \frac{{k{q_1}{q_2}}}{{{r^2}}}[/katex]
Substitute Q to [katex]{q_1}[/katex]
[katex]F = \frac{{k{q_1}{q_2}}}{{{r^2}}}[/katex]
The vector form for the force on charge B due charge D is
[katex]F = \frac{{k{q_1}{q_2}}}{{{r^2}}}[/katex]
Substitute [katex]F = \frac{{k{q_1}{q_2}}}{{{r^2}}}[/katex]
[katex]F = \frac{{k{q_1}{q_2}}}{{{r^2}}}[/katex]
The vector [katex]{\vec F_{{\rm{net,B}}}}[/katex] magnitude
[katex]{\vec F_{{\rm{net,B}}}}[/katex]
Ans:
The [katex]{\vec F_{{\rm{net,B}}}}[/katex]1 is the magnitude of the net force that acts on a charge when three identical charges are placed at each corner of a square.