Lost your password? Please enter your email address. You will receive a link and will create a new password via email.
Please briefly explain why you feel this question should be reported.
Please briefly explain why you feel this answer should be reported.
Please briefly explain why you feel this user should be reported.
Two small diameter 10gm dielectric balls can slide freely on a vertical channel. Each carry a negative charge of 1μC. Find the separation between the balls if the lower ball is restrained from moving
To find the separation between the two charged dielectric balls, we can employ Coulomb's Law and the equilibrium condition due to the gravitational force acting on the upper ball.Coulomb's Law gives us the electrostatic force (F_e) between two charges:[ F_e = k_e frac{|q_1 q_2|}{r^2} ]where- (k_e =Read more
To find the separation between the two charged dielectric balls, we can employ Coulomb’s Law and the equilibrium condition due to the gravitational force acting on the upper ball.
Coulomb’s Law gives us the electrostatic force (F_e) between two charges:
[ F_e = k_e frac{|q_1 q_2|}{r^2} ]
where
– (k_e = 8.99 times 10^9 , text{N}cdottext{m}^2/text{C}^2) is the Coulomb’s constant,
– (q_1) and (q_2) are the charges of the balls, which are each (1 mu C = 1 times 10^{-6} C),
– (r) is the separation between the centers of the two balls, which we are trying to find.
The gravitational force (F_g) acting on the upper ball is given by:
[ F_g = mg ]
where
– (m = 10 , text{gm} = 0.01 , text{kg}) is the mass of the ball,
– (g = 9.8 , text{m/s}^2) is the acceleration due to gravity.
At equilibrium, the electrostatic force of repulsion between the balls is equal to the gravitational force pulling the upper ball downwards:
[ F_e = F_g
See lessFind the force between two charges when they are brought in contact and separated by 4cm apart, charges are 2nC and -1nC, in μN
To find the force between two charges, we can use Coulomb's Law, which is expressed as:[F = k cdot frac{|q_1 cdot q_2|}{r^2}]where:- (F) is the force between the charges,- (k) is Coulomb's constant ((8.987 times 10^9 N m^2/C^2)),- (q_1) and (q_2) are the magnitudes of the two charges (in this case,Read more
To find the force between two charges, we can use Coulomb’s Law, which is expressed as:
[F = k cdot frac{|q_1 cdot q_2|}{r^2}]
where:
– (F) is the force between the charges,
– (k) is Coulomb’s constant ((8.987 times 10^9 N m^2/C^2)),
– (q_1) and (q_2) are the magnitudes of the two charges (in this case, 2nC and -1nC, which are (2 times 10^{-9}C) and (-1 times 10^{-9}C) respectively),
– and (r) is the distance between the charges (4cm, which needs to be converted to meters, so (r = 0.04 m)).
Substituting the given values:
[F = 8.987 times 10^9 N m^2/C^2 cdot frac{(2 times 10^{-9} C) cdot (1 times 10^{-9} C)}{(0.04 m)^2}]
[F = 8.987 times 10^9 cdot frac{2 times 10^{-18}}{0.0016}]
[F = 8.987 times 10^9 cdot 1.
See lessFind the force of interaction between 60 stat coulomb and 37.5 stat coulomb spaced 7.5cm apart in transformer oil(εr=2.2) in 10-4 N
To calculate the force of interaction between two charges in a medium other than vacuum, we can use Coulomb's Law, modified to account for the dielectric constant of the medium ((epsilon_r)).The formula for the force between two point charges in a vacuum is given by Coulomb's law as:[F = k frac{{|q_Read more
To calculate the force of interaction between two charges in a medium other than vacuum, we can use Coulomb’s Law, modified to account for the dielectric constant of the medium ((epsilon_r)).
The formula for the force between two point charges in a vacuum is given by Coulomb’s law as:
[
F = k frac{{|q_1 q_2|}}{{r^2}}
]
where
– (F) is the force between the charges,
– (q_1) and (q_2) are the magnitudes of the two charges,
– (r) is the distance between the charges, and
– (k) is Coulomb’s constant. For calculations in the cgs system (centimeter, gram, second), specifically for charges measured in statcoulombs, (k) is set to 1 in the appropriate units, which simplifies the equation to:
[
F = frac{{|q_1 q_2|}}{{r^2}}
]
When considering a medium with a dielectric constant (epsilon_r), the force is reduced compared to the force in a vacuum. The modified Coulomb’s law takes the dielectric constant into account:
[
F = frac{{1}}{{4 pi epsilon_0 epsilon_r}} frac{{|q_1 q_2|}}{{r^2}}
]
In the cgs system, (4 pi
See lessTwo charges 1C and -4C exists in air. What is the direction of force?
The direction of the force between two charges, such as 1C and -4C existing in air, can be determined using Coulomb's law, which states that like charges repel each other and unlike charges attract each other. Since one charge is positive (1C) and the other is negative (-4C), the force between themRead more
The direction of the force between two charges, such as 1C and -4C existing in air, can be determined using Coulomb’s law, which states that like charges repel each other and unlike charges attract each other. Since one charge is positive (1C) and the other is negative (-4C), the force between them would be attractive. Thus, the direction of the force on each charge would be towards the other charge. Specifically, the 1C charge would experience a force in the direction of the -4C charge, and similarly, the -4C charge would experience a force in the direction of the 1C charge.
See lessFind the force between 2C and -1C separated by a distance 1m in air(in newton).
To find the force between two charges, we use Coulomb's Law, given by the formula:[ F = k cdot frac{|q_1 cdot q_2|}{r^2} ]where (F) is the force between the charges, (k) is Coulomb's constant ((8.987 times 10^9 , text{Nm}^2/text{C}^2)), (q_1) and (q_2) are the magnitudes of the two charges, and (r)Read more
To find the force between two charges, we use Coulomb’s Law, given by the formula:
[ F = k cdot frac{|q_1 cdot q_2|}{r^2} ]
where (F) is the force between the charges, (k) is Coulomb’s constant ((8.987 times 10^9 , text{Nm}^2/text{C}^2)), (q_1) and (q_2) are the magnitudes of the two charges, and (r) is the distance between the charges.
Given:
– (q_1 = 2C)
– (q_2 = -1C) (the negative sign indicates the nature of the charge, which affects the direction of the force but not its magnitude, as we use the absolute value in Coulomb’s Law)
– (r = 1m)
[ F = 8.987 times 10^9 , text{Nm}^2/text{C}^2 cdot frac{|2 cdot -1|}{1^2} ]
[ F = 8.987 times 10^9 , text{Nm}^2/text{C}^2 cdot 2 ]
[ F = 17.974 times 10^9 , N ]
Therefore, the magnitude of the force between the
See lessFor a function given by F = 4x i + 7y j +z k, the divergence theorem evaluates to which of the values given, if the surface considered is a cone of radius 1/2π m and height 4π2 m.
To find the divergence of the vector field (mathbf{F} = 4x mathbf{i} + 7y mathbf{j} + z mathbf{k}) and apply the divergence theorem to the specific geometry given (a cone with radius (frac{1}{2pi}) m and height (4pi^2) m), we first calculate the divergence of (mathbf{F}).The divergence of a vector fRead more
To find the divergence of the vector field (mathbf{F} = 4x mathbf{i} + 7y mathbf{j} + z mathbf{k}) and apply the divergence theorem to the specific geometry given (a cone with radius (frac{1}{2pi}) m and height (4pi^2) m), we first calculate the divergence of (mathbf{F}).
The divergence of a vector field (mathbf{F} = Pmathbf{i} + Qmathbf{j} + Rmathbf{k}) is given by:
[ nabla cdot mathbf{F} = frac{partial P}{partial x} + frac{partial Q}{partial y} + frac{partial R}{partial z} ]
For (mathbf{F} = 4x mathbf{i} + 7y mathbf{j} + z mathbf{k}), we have:
[ nabla cdot mathbf{F} = frac{partial (4x)}{partial x} + frac{partial (7y)}{partial y} + frac{partial (z)}{partial z} = 4 + 7 + 1 = 12 ]
Now, the divergence theorem states that for a vector field (mathbf{F}
See lessIf a function is described by F = (3x + z, y2 − sin x2z, xz + yex5), then the divergence theorem value in the region 0<x<1, 0<y<3 and 0<z<2 will be
To solve for the divergence of a vector field (F = (3x + z, y^2 - sin(x^2z), xz + ye^{x^5})) and then use the divergence theorem to find the value in the specified region, we need to follow these steps: 1. Find the divergence of (F):The divergence of a vector field (F = (P, Q, R)) is given by:[ nablRead more
To solve for the divergence of a vector field (F = (3x + z, y^2 – sin(x^2z), xz + ye^{x^5})) and then use the divergence theorem to find the value in the specified region, we need to follow these steps:
1. Find the divergence of (F):
The divergence of a vector field (F = (P, Q, R)) is given by:
[ nabla cdot F = frac{partial P}{partial x} + frac{partial Q}{partial y} + frac{partial R}{partial z} ]
For (F = (3x + z, y^2 – sin(x^2z), xz + ye^{x^5})), we have:
– (P(x, y, z) = 3x + z)
– (Q(x, y, z) = y^2 – sin(x^2z))
– (R(x, y, z) = xz + ye^{x^5})
So,
[ frac{partial P}{partial x} = 3 ]
[ frac{partial Q}{partial y} = 2y ]
[ frac{partial R}{partial z} = x ]
Therefore, the divergence of (F) is:
[ nabla cdot F =
See lessThe divergence theorem value for the function x2 + y2 + z2 at a distance of one unit from the origin is
To apply the divergence theorem to the function (f(x, y, z) = x^2 + y^2 + z^2) at a distance of one unit from the origin, we first need to understand what the divergence theorem is and how it's used in this context.The divergence theorem, also known as Gauss's theorem, relates the flow (flux) of a vRead more
To apply the divergence theorem to the function (f(x, y, z) = x^2 + y^2 + z^2) at a distance of one unit from the origin, we first need to understand what the divergence theorem is and how it’s used in this context.
The divergence theorem, also known as Gauss’s theorem, relates the flow (flux) of a vector field through a closed surface to the divergence of the vector field inside the volume enclosed by the surface. Mathematically, it can be expressed as:
[
intint_{Sigma} vec{F} cdot dvec{A} = intintint_{V} nabla cdot vec{F} dV
]
where (Sigma) is the closed surface, (V) is the volume enclosed by (Sigma), (vec{F}) is a vector field, (dvec{A}) is a vector element of area on (Sigma), and (dV) is an element of volume inside (V).
For the function (f(x, y, z) = x^2 + y^2 + z^2), we first convert this scalar function into a vector field. A common way to do this, given the context is not explicitly directing towards a specific vector field construction, is to consider a gradient field derived from (f), since the divergence theorem
See lessFind the Gauss value for a position vector in Cartesian system from the origin to one unit in three dimensions.
When discussing the Gauss value related to a position vector in the context of magnetic or electric fields, this term doesn't directly correlate to a specific "Gauss value" for a position vector because Gauss's law and values typically pertain to the flux of electric or magnetic field through a surfRead more
When discussing the Gauss value related to a position vector in the context of magnetic or electric fields, this term doesn’t directly correlate to a specific “Gauss value” for a position vector because Gauss’s law and values typically pertain to the flux of electric or magnetic field through a surface and not directly to the characteristics of position vectors themselves. However, if we’re looking to understand a basic magnetic or electric field strength (in teslas or gauss, respectively, for magnetic fields) at a point in space due to a position vector from the origin to a point, more context or specifics about the sources of the field and their relations to the vector would be needed.
Without additional specifics—like the nature of the source of the magnetic or electric field, and whether you’re interested in fields generated by point charges, currents, or dipoles, or if you’re looking for an application of Gauss’s law (for electromagnetism) to a given configuration—it’s not possible to provide a numeric “Gauss value” for a position vector. Gauss’s law, in its essence for electricity, relates the electric flux through a closed surface to the charge enclosed by that surface, not directly assigning a value to a position vector.
For a magnetic field, the strength is often measured in Gauss or Tesla, where 1 Tesla = 10,000 Gauss. But the strength of the field depends on the specifics of the magnetic source and its distance from the point of interest, rather than just the existence
See lessEvaluate the surface integral ∫∫ (3x i + 2y j). dS, where S is the sphere given by x2 + y2 + z2 = 9
To evaluate the surface integral (iint (3xmathbf{i} + 2ymathbf{j}) cdot dmathbf{S}), where (S) is the sphere given by (x^2 + y^2 + z^2 = 9), we use the fact that the sphere has radius (r=3) and is centered at the origin.Given the vector field (mathbf{F} = 3xmathbf{i} + 2ymathbf{j}), notice that theRead more
To evaluate the surface integral (iint (3xmathbf{i} + 2ymathbf{j}) cdot dmathbf{S}), where (S) is the sphere given by (x^2 + y^2 + z^2 = 9), we use the fact that the sphere has radius (r=3) and is centered at the origin.
Given the vector field (mathbf{F} = 3xmathbf{i} + 2ymathbf{j}), notice that the vector field’s third component is zero (F_z=0), implying that it has no component in the (z)-direction.
The surface integral over a closed surface, like a sphere, can be computed via the divergence theorem. However, in this specific case, calculating the vector field’s dot product with the outward normal directly and integrating over the surface might not be straightforward due to the absence of the (z)-component in (mathbf{F}). Nonetheless, it is more insightful to exploit the symmetry of the sphere and the nature of the vector field for this calculation.
Symmetry Insight:
See less1. For the component (3xmathbf{i}), its effect cancels out symmetrically in the integral over the sphere because for every (x), there is a (-x) with equal contribution but opposite directions when projected to the surface area element (dmath