Download - Coulomb’s Law
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Coulomb’s Law• Coulomb’s Law• Electric Field• Electric Field from Multiple Charges• Integration of Volume charge• Electric Field near Infinite Wire• Electric Field near Infinite Sheet• Electric Field between two Infinite Sheets• Field Lines• Streamlines
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Coulomb’s Law• Coulomb’s Law with k = 9 x 109 Nm2/C2
εo= 8.85 x 10-12 C2 / Nm2
• Unit vector from r1 to r2
• Combining
• (Action reaction F1 = -F2)
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Example of Coulomb’s Law• Force of charge 1 on charge 2
– Charge 1 - 3 x 10-4 C at M(1,2,3)– Charge 2 - -1 x 10-4 C at N(2,0,5)
• Coulomb’s Law
• R magnitude
• Unit vector
• Result
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Electric Field• Electric Field
– Coulomb’s Law without 2nd charge– Separates Problem into “Background” and “Test Charge”– Units newtons/coulomb (volts/meter)
– For source charge at r’ observed at r’
• For source charges at r1and r2, observed at r
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Electric Field from Multiple Charges• 2 source charges at 1 and 2, observed at r
• Multiple source charges at m, observed at r
• Infinite # source charges, observed at r
• We’re going to spend some time on the last one!
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Example – Electric Field from 4 charges• Sources charges at P1(1,1,0), P2(-1,1,0), P3(-1,-1,0), P4(1,-1,0). Each 3 nC.
• Observation point r at P(1,1,1)
– P1(1,1,0)
– P2(-1,1,0)
– P3(-1,-1,0)
– P4(1,-1,0)
• Total field is:
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Continuous Charge - Integration of Charge
• Differential charge element
• Integrate for total charge
• Example – charge density – Find total charge over region 0 <ρ<1 cm, 2cm <z< 4cm
• Comments– Dependence on ρ and z in negative exponential causes rapid fall-off in ρV
– Concentrated near z= 0 plane where exponential is small– Concentrated near ρ = 0 z-axis where exponential is small
• Integral
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Integration of Charge (cont)• Integration on φ
• Integration on z
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Continuous Charge - Other examples• Setup Cartesian
– Integrate volume – Subtract volume 1– Q will be zero from integration of odd function.
• Setup cylindrical– )
– Differential volume
• Universe
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Continuous Charge -Middle Example• Integral is
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Continuous Charge -Field near infinite line charge
• Will do in cylindrical coordinates– Observation on y axis, z = 0 plane – Source distributed along z axis – Linear charge density constant
• Source to observation vector
• Differential Field Contribution
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Field near infinite line of charge (cont)• ρ and z components
(odd - integrates to zero)
• Integration for a long wire is thus
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Field near infinite sheet of charge• Given an infinite line charge and surface density ρs
• x and y components
(odd - integrates to zero)
(symmetry)
• x component
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Field near infinite sheet (cont)• Integration for a sheet is thus
• Field points away toward
• Field is independent of distance r<<width
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Electric Field between 2 Infinite Sheets
(-Q) charge sign and unit vector reversed)
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Field Lines• Field lines– Point in direction of electric field
– Direction + test charge moves
– Originates on +Q terminates on -Q
– Cross-sectional density proportional to E magnitude
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Streamlines• Equation of line which follows field line at x, y, z
– slope of this line y=f(x) – should equal field ratio
– Set
– Solve for equation y=f(x) as function of x
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Streamlines• Vector field are Ax, Ay, and Az function of x,y,z• From geometry
• Example
• Plugging in
• Result
• Plug in x and y at particular pointto evaluate C
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Streamline Example
• Find streamlines of following in rectangular coordinates
• Transforming to rectangular
• Plugging in streamline equation
• Solution
• At P(-2,7,10) y = -3.5 x
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Example problem 11. 3 point charges are in xy plane; with 5 nC at y= 5 cm, -10 nC at y =-5 cm, and 15 nC
at x=-5cm. Find position of 20 nC that exactly cancels field at origin.
- Add first 3 fields to get resultant as function of ax , ay (like example 2.2)
- 4th charge must exactly cancel field with same combination of ax , ay
- Write in general field form as magnitude times unit vector
- Equate magnitudes
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Example problem 27. A 2uC charge is located at A(4,3,5) in free space. Find Eρ, Eφ, and Ez at
P(8,12,2)- Get field in rectangular coordinates as function of ax, ay, az
- translate rectangular variables to cylindrical variables- translate rectangular unit vectors to cylindrical variables.
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Example problem 313. A uniform charge density extends throughout a spherical shell from r=3
cm to r=5 cm. Find the total charge and the radius containing half the charge.
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Example problem 4• Find the electric field on the z-axis produced byan annular ring z= 0, a <ρ
<b, 0 < φ < 2π