Consider two positive charges with q1 = 4 q2. The magnitude of the force on q1 is: A. Four times larger than the force on q2 B. Four times smaller than the force on q2 C. Two times... Consider two positive charges with q1 = 4 q2. The magnitude of the force on q1 is: A. Four times larger than the force on q2 B. Four times smaller than the force on q2 C. Two times larger than the force on q2 D. Two times smaller than the force on q2 E. Equal to the force on q2
Understand the Problem
The question is asking to calculate the magnitude of the force exerted on a charge q1 in relation to another charge q2, given that q1 is four times the value of q2. The context involves Coulomb's law, which describes the electrostatic force between two charges.
Answer
The magnitude of the force on \( q_1 \) is four times larger than the force on \( q_2 \).
Answer for screen readers
The magnitude of the force on ( q_1 ) is: $$ \text{Four times larger than the force on } q_2 $$
Steps to Solve
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Identify the Charges Given that ( q_1 = 4q_2 ), we can denote the magnitudes of the charges as:
- ( q_1 = 4 \cdot q_2 )
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Write the Equation for the Force Using Coulomb's law, the formula for the magnitude of the force ( |F| ) between the two charges is: $$ |F| = k \frac{|q_1||q_2|}{r^2} $$
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Substitute ( q_1 ) in the Equation Replace ( q_1 ) with ( 4q_2 ): $$ |F| = k \frac{|4q_2||q_2|}{r^2} $$
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Simplify the Expression This simplifies to: $$ |F| = k \frac{4|q_2|^2}{r^2} $$
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Determine the Force on ( q_2 ) Using the same formula for the force acting on ( q_2 ): $$ |F_{q_2}| = k \frac{|q_2||q_1|}{r^2} $$ Substituting ( q_1 = 4q_2 ): $$ |F_{q_2}| = k \frac{|q_2||4q_2|}{r^2} $$ This gives: $$ |F_{q_2}| = k \frac{4|q_2|^2}{r^2} $$
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Compare Forces Since:
- ( |F_{q_1}| = 4 |F_{q_2}| ) The force exerted on ( q_1 ) is four times the force on ( q_2 ).
The magnitude of the force on ( q_1 ) is: $$ \text{Four times larger than the force on } q_2 $$
More Information
According to Coulomb's law, the electrostatic force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This relationship leads us to the conclusion that if one charge is larger (in this case, four times larger), the force it experiences will also be proportionally larger.
Tips
- Confusing Forces: Sometimes, students confuse the magnitude of force acting on one charge with that acting on the other. Remember, the forces are equal in magnitude but act in opposite directions.
- Misapplying Coulomb's Law: Ensure that the values for charges are correctly plugged into the formula.
- Ignoring Proportional Relationships: Students may overlook how the ratios of the magnitudes of the charges affect the resulting forces.
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