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Electric Charges and Fields 02 || Coulomb's Law and Force Between Multiple Charges JEE MAINS/NEET

Physics Wallah - Alakh Pandey・2 minutes read

Kalm's Law explains the interaction of electric charges, stating that like charges repel and opposite charges attract, quantifying the force through the formula \( F = k \frac{q_1 q_2}{r^2} \); this force is influenced by the charges' magnitudes and distance, following the Inverse Square Law. Additionally, the lecture highlights the significance of understanding these principles for problem-solving in competitive exams like JEE and NEET, while encouraging students to engage actively with the material and practice related numerical problems.

Insights

  • Kalm's Law explains the behavior of electric charges, stating that like charges repel each other while opposite charges attract, with the force acting along the line connecting their centers.
  • The strength of the electrostatic force between two point charges is calculated using the formula \( F = k \frac{q_1 q_2}{r^2} \), where \( k \) varies depending on the medium, and this relationship highlights that the force is stronger when the charges are closer together and weaker when they are further apart.
  • Coulomb's constant \( k \) is approximately \( 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \) in a vacuum, but this value can change when charges are in different materials, such as water or air, affecting the overall force between the charges.
  • The concept of point charges is crucial for simplifying calculations in electrostatics, as it allows for the assumption that charges are concentrated in a very small space, making it easier to apply Kalm's Law without considering the complexities of charge distribution.
  • The principle of superposition is essential in electrostatics, as it states that the total force on a charge in a system of multiple charges is the vector sum of the individual forces exerted by all other charges, which is vital for solving problems involving more than two charges.
  • In scenarios involving equilibrium, the placement of a third charge between two others affects the net forces acting on it; for charges of the same type, the third charge should be positioned at the midpoint, while for charges of different magnitudes, it should be closer to the smaller charge to achieve balance.
  • Finally, the relationship between electrostatic forces and motion is emphasized through the concept of Simple Harmonic Motion (SHM), illustrating that when a charge is displaced from its equilibrium position, it experiences a restoring force that is proportional to the displacement, demonstrating oscillatory behavior.

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Recent questions

  • What is the definition of electrostatics?

    Electrostatics is the study of electric charges at rest. It involves understanding the forces, fields, and potentials associated with stationary electric charges. The fundamental principles of electrostatics are governed by Coulomb's Law, which describes the interaction between charged particles. Electrostatics plays a crucial role in various applications, including capacitors, insulators, and the behavior of materials in electric fields. The concepts of electric field strength, potential difference, and charge distribution are essential in analyzing electrostatic phenomena.

  • How do I calculate electric force?

    To calculate the electric force between two point charges, you can use Coulomb's Law, which states that the force \( F \) is proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The formula is given by \( F = k \frac{q_1 q_2}{r^2} \), where \( k \) is Coulomb's constant, approximately \( 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \), \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance separating them. This formula allows you to determine the magnitude and direction of the force acting between the charges, which is essential for understanding interactions in electrostatics.

  • What is a point charge?

    A point charge is an idealized model of an electric charge that is concentrated at a single point in space, with no physical size or volume. This simplification allows for easier calculations and analysis of electric fields and forces. In practice, point charges are used to represent small charged objects where the distance between charges is much larger than their sizes. The behavior of point charges is governed by Coulomb's Law, which describes how they interact with each other through electric forces. Understanding point charges is fundamental in electrostatics, as they serve as the building blocks for more complex charge distributions.

  • What is the principle of superposition?

    The principle of superposition in electrostatics states that the net force acting on a charge due to multiple other charges is the vector sum of the individual forces exerted by each charge. This means that when calculating the total force on a charge, you can analyze the effect of each charge separately and then combine these effects to find the overall force. This principle is crucial for solving problems involving multiple charges, as it allows for systematic calculations of forces in complex configurations. The superposition principle is widely applicable in various fields, including physics and engineering, where multiple influences must be considered simultaneously.

  • How do I find charge equilibrium?

    To find charge equilibrium, you need to determine the position of a third charge such that the net force acting on it from two other charges is zero. If the two charges are of opposite signs, the third charge should be placed outside the two charges, closer to the smaller magnitude charge. If both charges are of the same type, the third charge should be positioned exactly in the middle. The equilibrium condition can be mathematically expressed by setting the magnitudes of the forces acting on the third charge equal to each other. This involves using Coulomb's Law to calculate the forces and solving the resulting equation to find the appropriate position for the third charge, ensuring that it experiences no net force.

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Summary

00:00

Understanding Kalm's Law of Electric Charges

  • Kalm's Law describes the interaction of electric charges, stating that like charges repel and opposite charges attract, with the force acting along the line joining the centers of the charges.
  • The formula for the force between two point charges is given by \( F = k \frac{q_1 q_2}{r^2} \), where \( F \) is the force, \( k \) is a constant that depends on the medium, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between the charges.
  • The constant \( k \) for free space or vacuum is approximately \( 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \), and this value may vary in different mediums such as water or air.
  • The force is directly proportional to the product of the charges \( q_1 \) and \( q_2 \) and inversely proportional to the square of the distance \( r \) between them, illustrating the Inverse Square Law.
  • If the distance between two charges is doubled, the force of interaction decreases by a factor of four, while halving the distance increases the force by a factor of four.
  • Kalm's Law applies specifically to point charges, which are defined as charges concentrated in a very small space, allowing for the simplification of calculations.
  • The lecture emphasizes the importance of making notes, which will be provided in PDF format within seven days, and encourages students to engage with the material actively.
  • The concept of point charges is explained through an example of a large metallic ball, clarifying that the charge distribution must be considered in relation to the distance between charges to determine if they can be treated as point charges.
  • The lecture will continue with further discussions on the medium's effect on the force and the concept of permittivity in future sessions, indicating a structured approach to learning.
  • Students are encouraged to practice numerical problems related to Kalm's Law, with a focus on understanding the relationships between charge, distance, and force, as these concepts are frequently tested in exams like JEE and NEET.

15:29

Vector Forces and Unit Vectors Explained

  • To find the unit vector of a given vector \( \mathbf{a} \), divide the vector \( \mathbf{a} \) by its magnitude, expressed as \( \mathbf{u} = \frac{\mathbf{a}}{|\mathbf{a}|} \), where \( |\mathbf{a}| \) is the magnitude of vector \( \mathbf{a} \).
  • The formula for the electrostatic force between two charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by Coulomb's Law: \( F = k \frac{q_1 q_2}{r^2} \), where \( k \) is Coulomb's constant, approximately \( 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \).
  • The unit vector in the direction of the vector \( \mathbf{r_1} - \mathbf{r_2} \) can be calculated as \( \hat{\mathbf{r}} = \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|} \), where \( |\mathbf{r_1} - \mathbf{r_2}| \) is the magnitude of the vector difference.
  • The magnitude of the vector \( \mathbf{r_1} - \mathbf{r_2} \) is calculated using the formula \( |\mathbf{r_1} - \mathbf{r_2}| = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2} \).
  • The electrostatic force can also be expressed in vector form, where the direction is indicated by the unit vector \( \hat{\mathbf{r}} \): \( \mathbf{F} = k \frac{q_1 q_2}{|\mathbf{r_1} - \mathbf{r_2}|^2} \hat{\mathbf{r}} \).
  • The gravitational force between two masses \( m_1 \) and \( m_2 \) is given by \( F_g = G \frac{m_1 m_2}{r^2} \), where \( G \) is the gravitational constant, approximately \( 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \).
  • The ratio of the electrostatic force to the gravitational force for two electrons can be calculated using their respective formulas, yielding a result of approximately \( 10^{42} \), indicating that the electrostatic force is significantly stronger than the gravitational force.
  • The mass of an electron is approximately \( 9.11 \times 10^{-31} \, \text{kg} \) and the charge of an electron is \( 1.6 \times 10^{-19} \, \text{C} \).
  • To maximize the electrostatic force between two charges \( q_1 \) and \( q_2 \) divided into parts \( x \) and \( q - x \), set the derivative of the force equation with respect to \( x \) to zero, leading to the condition for maximum force.

32:07

Understanding Charge Interactions and Forces

  • The discussion begins with the concept of differentiability and the need to analyze the relationship between variables, specifically focusing on the cancellation of terms when k is lost, leading to the conclusion that the value of q equals 2x, which implies x = q/2.
  • It is emphasized that when dividing a charge to achieve maximum force, it should be split into two equal parts, as demonstrated with a mass m, where the gravitational force is maximized when m is divided into m/2 and m/2.
  • The limitations of Coulomb's Law are outlined, stating that it is not applicable to extended charges and is only valid for point charges, particularly when both charges are at rest.
  • The text clarifies that Coulomb's Law can only be applied when both charges are stationary; if either charge is in motion, the law does not hold.
  • The principle of superposition of charges is introduced, explaining that the net force on a charge in a system of multiple charges is the vector sum of the forces exerted by all other charges on it.
  • An example is provided with three charges (q1, q2, q3) to illustrate how to calculate the net force on a charge due to the presence of others, emphasizing the need for vector addition.
  • The net force on a charge is defined as the vector sum of the forces due to all remaining charges, which is crucial for solving problems involving multiple charges.
  • A numerical example involving an equilateral triangle with three equal charges is presented, where the forces acting on one charge are calculated, taking into account the angles between the forces.
  • The resultant force from two equal forces acting at a 60-degree angle is derived using vector addition, resulting in a net force magnitude of √3f at an angle of 30 degrees.
  • The text concludes with another example involving a right triangle configuration of charges, reiterating the importance of vector analysis in determining the net force on a charge due to the influence of others.

48:38

Vector Forces and Charge Interactions Explained

  • The discussion begins with vector physics, specifically focusing on the forces acting on charges arranged in a square configuration, with charges denoted as +q and -q at the corners and center of the square.
  • The principle of superposition is introduced, stating that the net force on a charge can be calculated by summing the individual forces exerted by other charges, emphasizing the need to consider both magnitude and direction.
  • The distance between charges is calculated using the formula for the diagonal of a square, which is a√2, where 'a' is the length of the side of the square.
  • The forces acting on charge C due to charges A and B are analyzed, with the forces being equal in magnitude and directed towards charge C, leading to the expression for the force as F = k(q1*q2)/r².
  • The resultant force from two equal forces acting at a 90° angle is derived using the Pythagorean theorem, resulting in a net force of F√2 at a 45° angle.
  • The net force on charge C is calculated by combining the forces from charges A, B, and D, with the final expression being F_net = k(q*q)/(a√2) + k(q*q)/(2a) after simplifying the terms.
  • A symmetrical charge distribution scenario is presented, where a charge placed at the center of a symmetrical arrangement experiences zero net force due to equal and opposite forces from surrounding charges.
  • The concept of net force being zero in symmetrical charge distributions is reinforced, with examples illustrating that any charge placed at the center of a symmetrical arrangement will not experience a net force.
  • A variation of the symmetrical charge problem is introduced, where an asymmetrical arrangement leads to a non-zero net force, prompting the calculation of forces based on their respective distances and magnitudes.
  • The lecture concludes with a reminder of the importance of consistency and practice in mastering the concepts of vector forces and charge interactions, encouraging students to engage with various problem types to solidify their understanding.

01:04:17

Charge Equilibrium in Electrostatics Explained

  • The concept of charge equilibrium involves determining the position of a third charge such that the net force acting on it is zero, given two other charges. This is essential for solving problems related to electrostatics.
  • When two charges are present, one positive and one negative, the third charge must be placed on the line joining the two charges to achieve equilibrium. The position can vary, but it must be strategically chosen to balance the forces.
  • If both charges are of the same type (either both positive or both negative), the third charge must be placed exactly in the middle of the two charges to ensure that the forces acting on it cancel each other out.
  • The third charge can be either positive or negative, but the placement remains the same; it should be positioned between the two charges. If the charges are the same, the equilibrium point is at the midpoint.
  • For two charges of different magnitudes, the third charge should be placed closer to the smaller charge. If the smaller charge is at a distance \( x \) from the third charge, the distance from the larger charge becomes \( L - x \).
  • The forces acting on the third charge can be calculated using Coulomb's law, where the force \( F_1 \) due to the first charge is given by \( F_1 = k \frac{q_1 q}{x^2} \) and the force \( F_2 \) due to the second charge is \( F_2 = k \frac{q_2 q}{(L - x)^2} \).
  • To find the equilibrium position, set the magnitudes of the forces equal: \( F_1 = F_2 \). This leads to the equation \( k \frac{q_1 q}{x^2} = k \frac{q_2 q}{(L - x)^2} \), which simplifies to \( \frac{q_1}{x^2} = \frac{q_2}{(L - x)^2} \).
  • Solving the equation yields the position of the third charge. For example, if \( q_1 = 4q \) and \( q_2 = q \), the solution will show that the third charge should be placed at \( \frac{L}{3} \) from the smaller charge.
  • In cases where one charge is positive and the other is negative, the third charge should be placed outside the two charges, closer to the smaller magnitude charge, to achieve equilibrium.
  • The equilibrium condition can be verified by ensuring that the forces acting on the third charge from both other charges are equal in magnitude but opposite in direction, leading to a net force of zero.

01:19:03

Electrostatic Forces and Simple Harmonic Motion

  • The force between two charges is given by the formula \( F = k \cdot \frac{9q \cdot q_2}{r^2} \), where \( k \) is the electrostatic constant, \( q \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between them.
  • To find the distance \( l + x \) from a reference point, the equation \( l + x = 3x \) can be rearranged to yield \( l + x = -3 \), indicating that \( l + x \) is negative, which suggests an error in the initial assumptions about the direction of \( x \).
  • The value of \( x \) can be calculated as \( x = -\frac{l}{4} \), indicating that the charge is positioned at a distance of \( l - \frac{l}{4} \) from the reference point, which is \( \frac{3l}{4} \).
  • When dealing with opposite charges, the third charge must be placed outside the two charges, and the distance \( x \) can be calculated using the formula \( x = \sqrt{\frac{q_1 \cdot q_2}{k}} \), where \( q_1 \) is the smaller charge and \( q_2 \) is the larger charge.
  • The formula for the equilibrium position of a charge between two others is derived from the principle of superposition, where the net force on the charge must equal zero, leading to the equation \( F_1 = F_2 \).
  • The time period of simple harmonic motion (SHM) can be determined using the formula \( T = 2\pi\sqrt{\frac{m}{k}} \), where \( m \) is the mass of the oscillating charge and \( k \) is the spring constant equivalent in the electrostatic context.
  • In equilibrium, if a charge is displaced slightly, the forces acting on it will cause it to oscillate back to its original position, demonstrating SHM characteristics.
  • The net force on a charge after a small displacement \( x \) can be expressed as \( F_{net} = k \cdot q \cdot \frac{4x}{a^2} \), where \( a \) is the distance between the charges and \( k \) is the electrostatic constant.
  • The relationship between the net force and displacement indicates that the force acts in the opposite direction to the displacement, confirming the nature of restoring forces in SHM.
  • The calculations and principles discussed are applicable to competitive exams like JEE and NEET, emphasizing the importance of understanding electrostatic forces and their implications in motion and equilibrium scenarios.

01:34:19

Understanding Simple Harmonic Motion Principles

  • The proof of Simple Harmonic Motion (SHM) requires demonstrating that the net force \( f \) is equal to \(-k\), where \( k \) is the spring constant, indicating that the net force and displacement are in opposite directions. The condition for SHM is established when the net force acting on a mass \( m \) is proportional to the negative of its displacement, leading to the equation \( f = -k \).
  • The time period \( T \) of SHM is calculated using the formula \( T = 2\pi \sqrt{\frac{m}{k}} \), where \( m \) is the mass of the particle undergoing SHM. In this context, if \( k \) is set to \( \frac{4}{a} \), the time period can be expressed as \( T = 2\pi \sqrt{\frac{m}{\frac{4}{a}}} \), simplifying to \( T = 2\pi \sqrt{\frac{ma}{4}} \).
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