Two particles are in a uniform electric field whose value is +2500 N / C. The mass and charge o

Two particles are in a uniform electric field whose value is...

Two particles are in a uniform electric field whose value is $+2500 \mathrm{~N} / \mathrm{C}$. The mass and charge of particle 1 are $m_{1}=1.4 \times 10^{-5} \mathrm{~kg}$ and $q_{1}=-7.0 \mu \mathrm{C},$ while the corresponding values for particle 2 are $m_{2}=2.6 \times 10^{-5} \mathrm{~kg}$ and $q_{2}=+18 \mu \mathrm{C} .$ Initially the particles are at rest. The particles are both located on the same electric field line, but are separated from each other by a distance $d$. When released, they accelerate, but always remain at this same distance from each other. Find $d$.

Key Concepts

Uniform Electric Field

A uniform electric field is characterized by a constant magnitude and direction at every point in space. This implies that a charged particle placed anywhere in the field experiences the same electric force per unit charge. Uniform fields are often used in problems to simplify analysis, as the force on each charge is given by F = qE, where q is the charge and E is the field strength.

Electric Force on a Charged Particle

When a charged particle is placed in an electric field, it experiences an electric force equal to the product of its charge and the electric field. This force, given by F = qE, acts in the direction of the field for positive charges and in the opposite direction for negative charges. It is the primary mechanism by which electric fields cause particles to accelerate.

Newton’s Second Law in the Context of Electrical Forces

Newton’s Second Law relates the net force acting on an object to its mass and the resulting acceleration (F = ma). When applied to charged particles in an electric field, the law is used to determine the acceleration due to both the external electric force and any additional forces, such as mutual forces between particles. This concept is crucial for analyzing the motion of the particles and ensuring their accelerations are compatible with a given condition, such as a constant separation.

Coulomb’s Law and the Mutual Force between Charges

Coulomb’s Law describes the electrostatic force between two point charges. The magnitude of the force is given by F = k|q1q2|/d², where k is Coulomb’s constant, q1 and q2 are the charges, and d is the distance separating them. This law accounts for the attractive nature of opposite charges and the repulsive nature of like charges, and it plays a central role when considering interactions between multiple charged particles.

Relative Equilibrium and Equal Acceleration Condition

For two particles to remain at a constant distance from each other while both accelerate, they must have the same acceleration. This condition, known as relative equilibrium in the accelerating frame, requires that any difference in acceleration due to external forces (like the uniform electric field) must be exactly balanced by the mutual forces (such as the Coulomb force). This concept is used to set up equations that relate the external and inter-particle forces so that the relative motion between the particles is zero.

Transcript

00:01 In this problem, we have two masses, each charged, one with a negative charge, one with a positive charge.

00:08 They are separated by distance d, and when they are placed in a stern electric field, they both accelerate, but this distance separating them, d does not change, which tells me immediately that each one shares the exact same acceleration.

00:28 Same magnitude, same direction.

00:30 It has to be.

00:31 Otherwise, they would part.

00:33 A d would change.

00:35 Now, this line here is the electric field line.

00:39 The electric field at any point is drawn tangent to it, following the general sense of the arrows on the field line.

00:47 So that means our electric field at points where m1 and m2r is to the right.

00:54 And with that said, let us look at forces on both charges, both masses.

01:04 So here it's going to be for m1.

01:06 Here's going to be for m2.

01:08 Minus plus means two puts an attractive force on 1.

01:14 So it's going to be f12.

01:17 1 puts the attractive force on 2.

01:21 Same magnitude, opposite direction.

01:26 That's given by quillum's law.

01:28 Now, the electric field, remember the definition.

01:33 It gives you the direction of the force on a positive charge.

01:37 So the electric field is pointing to the right at any of these places.

01:42 So on negative charge that are pointing to the right for the force, it'll point to the left, the opposite direction.

01:50 So this is what i'll call 1e.

01:55 And on m2, charge is positive, so it will point in the same direction as the electric field, which is to the right.

02:03 I'll call this 2e.

02:06 So now we have our forces.

02:09 We want to start writing out our equations.

02:17 So for m1, f -net x, f -1 -2 is in positive x, minus f -1e is equal to m -1 -a -x.

02:34 Remember, same acceleration for both.

02:36 There's no reason to subscript of 1 -x, a1 -x, a -2 -x, so all the same, a -x.

02:43 This i will call equation a -2 for m -2.

02:51 F -net -x is going to be f2e minus f -21 is equal to m2, a -x, this equation b.

03:07 Now, our goal at this point is to get a -x, and we know that f -1 -2, f -2 -1 are the same.

03:17 So let's eliminate those...

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