Coulomb's Law (Electric Force): Videos & Practice Problems

Coulomb's law describes the electric force between two charges, which can be attractive or repulsive based on their nature. The formula is given by $F^e=K{Q}^1{Q}^2/r^2$, where K is Coulomb's constant (8.99 x 10⁹ N m²/C²). The direction of the force is along the line connecting the charges, with like charges repelling and unlike charges attracting. Understanding this principle is crucial for analyzing electric interactions in various contexts.

Coulomb's Law

Coloumb's Law is a fundamental principle in electricity that describes the electric force between two charges. This law states that electric forces can either be attractive or repulsive, depending on the nature of the charges involved. Specifically, unlike charges (positive and negative) attract each other, while like charges (positive and positive or negative and negative) repel each other.

The mathematical expression for Coulomb's Law is given by:

$$ F = k \frac{Q_1 Q_2}{r^2} $$

where:

• F is the magnitude of the electric force between the two charges.
• k is Coulomb's constant, approximately equal to \( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \).
• Q_1 and Q_2 are the magnitudes of the two charges.
• r is the distance between the centers of the two charges.

This equation is analogous to the gravitational force equation, where the gravitational force is proportional to the product of the masses and inversely proportional to the square of the distance between them.

The direction of the force is along the line connecting the two charges. For repulsive forces, the direction is away from each other, while for attractive forces, it is towards each other. When calculating the magnitude of the force, it is common practice to use positive values for the charges and determine the direction based on the nature of the charges afterward.

To illustrate the application of Coulomb's Law, consider the example of a hydrogen atom, which consists of a proton and an electron. The electric force between these two particles can be calculated using their respective charges and the distance between them. The charge of a proton is \( +e \) (approximately \( 1.6 \times 10^{-19} \, \text{C} \)), and the charge of an electron is \( -e \). The distance between them is about \( 5.3 \times 10^{-11} \, \text{m} \). By substituting these values into Coulomb's Law, one can find the electric force acting on the hydrogen atom.

In addition to electric forces, gravitational forces can also be calculated for the same particles using Newton's law of gravitation:

$$ F_g = G \frac{m_1 m_2}{r^2} $$

where:

• F_g is the gravitational force.
• G is the gravitational constant, approximately \( 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \).
• m_1 and m_2 are the masses of the two particles.
• r is the distance between the centers of the two masses.

By calculating both forces, one can determine the ratio of the electric force to the gravitational force, revealing that the electric force is significantly stronger than the gravitational force in atomic interactions.

Another practical application of Coulomb's Law involves two identical charges connected by a wire. If the tension in the wire is known, it can be equated to the electric force exerted by the charges. By rearranging Coulomb's Law and substituting the known values, one can solve for the magnitude of the charges. This approach highlights the relationship between electric forces and mechanical tension in a system.

In summary, understanding Coulomb's Law is crucial for analyzing electric forces in various physical scenarios, from atomic structures to macroscopic systems. Mastery of this concept allows for deeper insights into the behavior of charged particles and their interactions.

If the force between two charges is F when the distance is d, what will the force between the two charges be if they were moved to a distance of 2d?

• A. 2F
• B. F/2
• C. F/4
• D. 4F

Answer: (C)

Charges In A Line (Find Zero Force)

To determine the position of a one Coulomb charge such that the net force acting on it is zero, we need to analyze the forces exerted by two other charges, one of 2 Coulombs and the other of 3 Coulombs. Since all charges are positive, they will repel each other, meaning the forces will always act in the direction away from each charge.

First, we can qualitatively assess the situation. If we place the one Coulomb charge to the left of the 2 Coulomb charge, the force from the 2 Coulomb charge will push it to the left, while the force from the 3 Coulomb charge will also push it to the left. Therefore, in this position, the forces cannot cancel each other out.

Similarly, if we place the one Coulomb charge to the right of the 3 Coulomb charge, the force from the 3 Coulomb charge will push it to the right, and the force from the 2 Coulomb charge will also push it to the right. Again, the forces will not cancel out in this scenario.

Thus, the one Coulomb charge must be positioned between the two other charges. The forces from the 2 Coulomb and 3 Coulomb charges will then act in opposite directions. However, since the 3 Coulomb charge is stronger, the one Coulomb charge must be closer to the 2 Coulomb charge to achieve equilibrium.

To find the exact position, we denote the distance from the one Coulomb charge to the 2 Coulomb charge as \( x \), and the distance from the one Coulomb charge to the 3 Coulomb charge as \( R - x \), where \( R \) is the total distance between the 2 Coulomb and 3 Coulomb charges (10 cm or 0.1 m).

The forces acting on the one Coulomb charge can be expressed using Coulomb's law, which states that the force \( F \) between two charges is given by:

\[ F = k \frac{q_1 q_2}{r^2} \]

where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between them.

For the force due to the 2 Coulomb charge:

\[ F_{2} = k \frac{2 \cdot 1}{x^2} \]

For the force due to the 3 Coulomb charge:

\[ F_{3} = k \frac{3 \cdot 1}{(R - x)^2} \]

Setting these two forces equal to each other for equilibrium gives:

\[ k \frac{2}{x^2} = k \frac{3}{(R - x)^2} \]

We can cancel \( k \) from both sides, leading to:

\[ \frac{2}{x^2} = \frac{3}{(R - x)^2} \]

Cross-multiplying yields:

\[ 2(R - x)^2 = 3x^2 \]

Expanding and rearranging gives:

\[ 2(R^2 - 2Rx + x^2) = 3x^2 \]

\[ 2R^2 - 4Rx + 2x^2 = 3x^2 \]

\[ 2R^2 - 4Rx - x^2 = 0 \]

This is a quadratic equation in terms of \( x \). Solving for \( x \) using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \( a = -1 \), \( b = -4R \), and \( c = 2R^2 \).

Substituting \( R = 0.1 \) m into the equation, we can find the value of \( x \). After calculations, we find:

\[ x \approx 0.045 \text{ m} \text{ or } 4.5 \text{ cm} \]

This means the one Coulomb charge should be placed approximately 4.5 cm from the 2 Coulomb charge, confirming that it is closer to the weaker charge to balance the forces effectively. The remaining distance to the 3 Coulomb charge would then be 5.5 cm.

In which direction will the −1 C charge move? If it has a mass of 10 g, what will its initial acceleration be?

• A. left; 3.8×10¹⁴ m/s²
• B. left; 6.2×10¹³ m/s²
• C. left; 6.2×10⁹ m/s²
• D. right; 1.1×10¹⁵ m/s²

Answer: (B)

Charges In A Triangle (Rank Force Pairs)

In this problem, we analyze the electric forces between pairs of point charges using Coulomb's Law, which states that the electric force \( F \) between two point charges is given by the formula:

\( F = k \frac{Q_1 Q_2}{r^2} \)

where \( k \) is Coulomb's constant, \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges, and \( r \) is the distance between them. In this scenario, the distance \( r \) is constant for all pairs, allowing us to focus solely on the product of the charges to determine the strength of the electric force.

We have three pairs of charges to consider:

• Pair 1: \( Q_1 = 2e \) and \( Q_2 = e \)
• Pair 2: \( Q_1 = 2e \) and \( Q_2 = 3e \)
• Pair 3: \( Q_1 = e \) and \( Q_2 = 3e \)

Calculating the products of the charges for each pair:

• For Pair 1: \( Q_1 Q_2 = (2e)(e) = 2e^2 \)
• For Pair 2: \( Q_1 Q_2 = (2e)(3e) = 6e^2 \)
• For Pair 3: \( Q_1 Q_2 = (e)(3e) = 3e^2 \)

Now, we can rank the pairs based on the magnitude of their charge products:

• Pair 2 has the greatest product: \( 6e^2 \)
• Pair 3 follows with \( 3e^2 \)
• Pair 1 has the smallest product: \( 2e^2 \)

Thus, the ranking of the pairs from greatest to least electric force is:

1. Pair 2: \( 2e \) and \( 3e \)
2. Pair 3: \( e \) and \( 3e \)
3. Pair 1: \( 2e \) and \( e \)

This analysis highlights the importance of charge magnitude in determining the strength of electric forces between point charges, as the distance remains constant in this scenario.

Charges in a Plane

In this scenario, we are analyzing the net force acting on a three coulomb charge positioned in a triangular arrangement with two other point charges: a negative two coulomb charge and a positive one coulomb charge. The forces acting on the three coulomb charge arise from the interactions with these two charges, leading to an attractive force from the negative charge and a repulsive force from the positive charge.

To determine the net force, we first identify the forces involved. The attractive force \( F_2 \) from the negative two coulomb charge is directed towards it, while the repulsive force \( F_1 \) from the positive one coulomb charge is directed away from it. The next step involves calculating these forces using Coulomb's Law, which is expressed as:

\( F = k \frac{|Q_1 Q_2|}{r^2} \)

where \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \), \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges, and \( r \) is the distance between them.

For \( F_2 \), the calculation involves:

\( F_2 = k \frac{(2 \, \text{C})(3 \, \text{C})}{(0.6 \, \text{m})^2} = 1.5 \times 10^{13} \, \text{N} \

For \( F_1 \), we first determine the distance between the three coulomb charge and the one coulomb charge using the Pythagorean theorem, yielding a distance of \( 0.1 \, \text{m} \). The force \( F_1 \) is then calculated as:

\( F_1 = k \frac{(3 \, \text{C})(1 \, \text{C})}{(0.1 \, \text{m})^2} = 2.7 \times 10^{12} \, \text{N} \

Next, we decompose these forces into their respective x and y components. The angle \( \theta \) in the triangle formed by the charges can be determined using trigonometric relationships. The cosine and sine of the angle are given by:

\( \cos(\theta) = \frac{6}{10} \quad \text{and} \quad \sin(\theta) = \frac{8}{10} \

Using these relationships, we can find the x and y components of \( F_1 \):

\( F_{1x} = F_1 \cos(\theta) = 2.7 \times 10^{12} \times \frac{6}{10} = 1.62 \times 10^{12} \, \text{N} \

\( F_{1y} = F_1 \sin(\theta) = 2.7 \times 10^{12} \times \frac{8}{10} = 2.16 \times 10^{12} \, \text{N} \

With the components calculated, we assign positive directions for upward and rightward forces, while leftward and downward forces are negative. The net force components can now be summed:

For the x-direction:

\( F_{net,x} = -1.5 \times 10^{13} + 1.62 \times 10^{12} = -1.34 \times 10^{13} \, \text{N} \

For the y-direction:

\( F_{net,y} = 2.16 \times 10^{12} \, \text{N} \

Finally, the magnitude of the net force is calculated using the Pythagorean theorem:

\( F_{net} = \sqrt{(-1.34 \times 10^{13})^2 + (2.16 \times 10^{12})^2} = 1.36 \times 10^{13} \, \text{N} \

This comprehensive approach illustrates how to analyze forces in a system of point charges, applying Coulomb's Law and vector decomposition to find the net force acting on a charge.

Exploiting Symmetry

Exploiting symmetry is a powerful technique in physics, particularly when analyzing the forces acting on charged particles. When dealing with arrangements of charges, such as two equal positive charges positioned symmetrically around a single charge, one can simplify the problem significantly by recognizing that the forces exerted by the charges will have equal magnitudes and symmetrical angles.

For instance, consider a scenario with two positive charges, each of +2 Coulombs, located at equal distances from a single +1 Coulomb charge. The forces acting on the +1 Coulomb charge from both +2 Coulomb charges will be repulsive. Since the distances are the same and the charges are identical, the magnitudes of the forces will also be equal. By decomposing these forces into their x and y components using trigonometric functions, one can observe that the y-components will cancel each other out due to symmetry, while the x-components will add together. This results in a net force that points directly upwards, equal to the sum of the y-components of the two forces.

In another example, if one charge is positive and the other is negative, the situation changes slightly. The positive charge will exert a repulsive force, while the negative charge will exert an attractive force on the same +1 Coulomb charge. Despite the different nature of the forces, the same principles of symmetry apply. The forces will still have equal magnitudes due to the same distance and charge values, and the angles will remain consistent. When decomposing these forces, the y-components will again cancel out, leaving a net force that is directed purely along the x-axis, which can be calculated as twice the x-component of either force.

Recognizing and utilizing symmetry in problems involving electric forces can greatly simplify calculations and enhance understanding of the net forces acting on charged particles. This approach is particularly useful in more complex arrangements, such as those involving multiple charges in geometric configurations like squares or circles, where symmetry can lead to straightforward solutions for the directions of net forces.

What is the direction of the net force on the charge at the center of the square in the following figure?

• A. Right
• B. Left
• C. Down
• D. The net force is 0 so its direction is undefined

Answer: (B)

Electroscope (Find Charge)

In this example, we explore the functioning of an electroscope, a device that demonstrates the principles of electrostatics. The electroscope consists of two conducting leaves that, when charged, repel each other due to their like charges. This repulsion causes the leaves to separate, and the angle of deflection can be measured. In this scenario, we are tasked with determining the charge on each leaf, given their mass and the deflection angle of 30 degrees.

Since both leaves carry identical charges, we can denote the charge on each leaf as \( Q \). The electric force \( F_e \) acting between the two charges can be described using Coulomb's Law, which states:

$$ F_e = \frac{k Q^2}{r^2} $$

Here, \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \), and \( r \) is the distance between the charges. Since both charges are equal, we can simplify the equation to:

$$ F_e = \frac{k Q^2}{r^2} $$

To find \( Q \), we rearrange the equation:

$$ Q = \sqrt{\frac{F_e r^2}{k}} $$

Next, we need to determine the distance \( r \) between the leaves and the electric force \( F_e \). The geometry of the situation reveals that the deflection angle creates an equilateral triangle, allowing us to calculate \( r \) based on the angle and the length of the strings holding the leaves. Given that the angle between the leaves is 60 degrees, we can deduce that the distance \( r \) is half the length of the string, which we assume to be 0.5 m.

To find the electric force \( F_e \), we analyze the forces acting on each leaf using a free body diagram. The leaves are in equilibrium, meaning the sum of forces in both the x and y directions must equal zero. The tension in the string provides a balancing force against the electric force and the weight of the leaves.

In the x-direction, we have:

$$ F_e - T \cos(\theta) = 0 $$

In the y-direction, we have:

$$ T \sin(\theta) - mg = 0 $$

From the y-direction equation, we can express the tension \( T \) as:

$$ T = \frac{mg}{\sin(\theta)} $$

Substituting this into the x-direction equation allows us to solve for \( F_e \). Given the mass of each leaf is 50 g (or 0.05 kg), we can calculate the weight \( mg \) and subsequently find \( T \) and \( F_e \). After calculating, we find \( T \) to be approximately 0.57 N, leading to an electric force \( F_e \) of about 0.29 N.

Now, substituting \( F_e \) and \( r \) back into our equation for \( Q \), we have:

$$ Q = \sqrt{\frac{0.29 \, \text{N} \cdot (0.5 \, \text{m})^2}{8.99 \times 10^9 \, \text{N m}^2/\text{C}^2}} $$

Upon solving, we find that the charge \( Q \) on each leaf is approximately \( 2.84 \times 10^{-6} \, \text{C} \), which can also be expressed as 2.84 microcoulombs (µC). This example illustrates the interplay between electric forces, geometry, and equilibrium in electrostatic systems.

Here’s what students ask on this topic:

What is Coulomb's Law and how is it used to calculate the electric force between two charges?

Coulomb's Law describes the electric force between two point charges. The formula is given by:

$F_e=K{Q}_1{Q}_2/r^2$

where $K$ is Coulomb's constant (8.99 x 10⁹ N m²/C²), $Q_1$ and $Q_2$ are the magnitudes of the charges, and $r$ is the distance between them. The force can be attractive or repulsive depending on the nature of the charges: like charges repel, unlike charges attract. This law is crucial for understanding electric interactions in various contexts, such as in atomic structures and electrical circuits.

How does the electric force between two charges compare to the gravitational force between two masses?

The electric force between two charges, described by Coulomb's Law, is generally much stronger than the gravitational force between two masses, described by Newton's Law of Gravitation. The formula for the electric force is:

$F_e=K{Q}_1{Q}_2/r^2$

while the gravitational force is given by:

$F_g=G{m}_1{m}_2/r^2$

where $G$ is the gravitational constant (6.67 x 10^-11 N m²/kg²). In a hydrogen atom, for example, the electric force is approximately 10³⁹ times stronger than the gravitational force, highlighting the dominance of electric interactions at the atomic scale.

What is Coulomb's constant and why is it important?

Coulomb's constant, denoted as $K$, is a proportionality factor in Coulomb's Law that quantifies the electric force between two point charges. Its value is approximately 8.99 x 10⁹ N m²/C². This constant is crucial because it allows us to calculate the magnitude of the electric force between charges accurately. Understanding and memorizing this constant is important for solving problems related to electric forces in physics, as it often appears in equations and calculations involving electric interactions.

How do you determine the direction of the electric force between two charges?

The direction of the electric force between two charges depends on whether the charges are like or unlike. Like charges (both positive or both negative) repel each other, so the force vectors point away from each other. Unlike charges (one positive and one negative) attract each other, so the force vectors point towards each other. The force always acts along the line connecting the two charges. To determine the direction, first calculate the magnitude of the force using Coulomb's Law, then apply the rules of attraction and repulsion to find the direction.

What is the relationship between the distance between two charges and the electric force between them?

The electric force between two charges is inversely proportional to the square of the distance between them, as described by Coulomb's Law:

$F_e=K{Q}_1{Q}_2/r^2$

As the distance $r$ between the charges increases, the force decreases rapidly. For example, if the distance is doubled, the force is reduced to one-fourth of its original value. This inverse-square relationship is crucial for understanding how electric forces behave over different distances.