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UF PHY 2049 - Electric Field

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PHY2049 Physics 2 Lecture Notes Electric Field D. Acosta Page 1 1/5/2008 Electric Fields Disclaimer: These lecture notes are not meant to replace the course textbook. The content may be incomplete. Some topics may be unclear. These notes are only meant to be a study aid and a supplement to your own notes. Please report any inaccuracies to the professor. Fields How does one charged particle attract or repel another particle at a distance? ⇒ Action at a distance For example, how do the electrons in a cell phone start moving due to a call emanating from a cell phone tower some distance away? In physics, we say that the force exerted by one object onto another a distance away is conveyed through a field. Electric Field The electric field is defined as the force acting on a positive test charge, per unit charge. 00 0 points in direction of qq≡>FEE F Units are thus N/C for the electric field. It is similar to the gravitational field on the surface of the Earth for a test mass m0 : 0m=Fg The electric field is a vector field. It has a magnitude and a direction. Another example vector field is the wind distribution. Temperature distribution is an example of a scalar field. 2 1 ?PHY2049 Physics 2 Lecture Notes Electric Field D. Acosta Page 2 1/5/2008 Field Lines Consider a positive test charge immersed in the electric field of a positive charge. • Lines indicate the direction of F if a test charge is placed there • Field lines point away from positive charge, and toward negative charge • Density of lines indicates the strength of the field • Field lines cannot cross. If they did, there would be more than one possible direction for a particle to travel at a given point. • The lines themselves have no meaning. The field is present everywhere. The magnitude of the electric field a distance r away from a point charge q: 2001 ; 4qKKqrπε== =FE i.e. dropped the q0 from Coulomb’s Law Electric Field of Several Point Charges Apply the superposition principle. This principle states that the resulting electric field is the sum of all fields, without any interference of one field upon another. It is generally true for electromagnetism at least for fields that are not enormously strong. For example, the total electric field at some point A from N charges is: 2111ˆNNiAi iAiiAqKr====∑∑EE r In other words, the fields from each point particle i a distance riA away from point A all add together, without any field affecting another. q0 +PHY2049 Physics 2 Lecture Notes Electric Field D. Acosta Page 3 1/5/2008 Electric Dipole Let’s consider the simplest multi-charge system: just two oppositely charged particles—a system referred to as an electric dipole. Let’s calculate the electric field anywhere along the x axis (result will actually apply anywhere in the x-z plane for y=0). y Let the top particle have charge q, And the bottom particle charge −q. The separation of charges is d. x By symmetry, E must point in the ˆ−y direction. Apply the superposition principle to determine it, by adding the electric field from each charge separately to get the net electric field. Consider a point along the x axis. The distance from either charge is ()22/2rxd=+ The magnitude of the electric field from each charge separately is ()222/2qqKKrxd+−== =+EE But the vector force points in a direction away from the positive charge and toward the negative one. i.e. ˆˆsin cos ˆˆsin cos θθθθ++ +−− −=−=− −EE xEyEE xEy So we see explicitly how the x-component of the total field cancels, and we are left with: TOTˆ2cos θ+− +=+=−EEE E y since +−=EE Now, ()22/2 /2cos/2ddrxdθ==+ So, y +q θ r d x −q E− E+ Fig. from HRW 7/ePHY2049 Physics 2 Lecture Notes Electric Field D. Acosta Page 4 1/5/2008 ()()() ()TOT2222TOT3/2 3/23222/2ˆ2/2/21 /2 1 /2qdKxdxdKqd Kqdxxd dx⎡⎤⎡⎤⎢⎥=−⎢⎥⎢⎥+⎢⎥+⎣⎦⎣⎦⇒= =⎡⎤ ⎡⎤++⎣⎦ ⎣⎦EyE By the binomial expansion, ()11nxnx+≈++L for small x. ()3/22TOT331 / 2 1 1 for 22ddx x dxqdKx−⎡⎤⇒+ ≈− ≈ >>⎣⎦⇒≈E If we had calculated the field along the y-axis, we’d have gotten: TOT32qdKy≈E Let’s define the electric dipole moment pqd≡ TOT3pKr⇒ E  where we have generalized from anywhere along the x axis to anywhere in the x-z plane, a distance r away from the dipole. Note that this field falls off faster than a single point charge, 2r−, because the opposite-sign charges contribute to the canceling of the electric field. This is a general behavior for the dipole field strength in any direction. For example, along the y axis, one gets a dipole field 2 times larger than the expression above (but all other terms the same). Continuous Charge Distributions Consider the electric field arising from an infinitesimal charge dq at a point a distance r away: 22ˆˆqKrdqdKrΔΔ=→=ErEr y r ΔE Δq xPHY2049 Physics 2 Lecture Notes Electric Field D. Acosta Page 5 1/5/2008 Projecting along each axis yields the following components of the infinitesimal electric field: ˆˆxydE ddE d=⋅=⋅ExEy Field from a Continuous Line Charge Now consider electric charge distributed uniformly along a 1-dimensional line from −L to L along the z-axis. The charge per unit length is λ (units: C/m) In terms of the total charge q, 2qLλ= So an infinitesimal length of the line, dz, has a charge dq = λdz. Let’s calculate the electric field along the y-axis (any axis perpendicular to the line will do). By symmetry, only the ˆycomponent of the field (that is, a field pointing perpendicular to the line) can survive when we integrate the field contributions from each infinitesimal charge dq along the length. 22; cosyyydq dqdK dEKrrEdEθ===∫E yxdqdq'dEdE’rθz+L-LPHY2049 Physics 2 Lecture Notes Electric Field D. Acosta Page 6 1/5/2008 222222coswhere cosyydqdE Krryzyyzθθ== ==+=+∫∫EE ()()22223/2221/2222221 by integral tables2LLLLLLdz yKyzyzdzKyyzzKyyyzKLyyLλλλλ−−−⇒= ⋅++=+=+=+∫∫E Consider yL<< , alternatively that L →∞ Then: where y can be replaced by the distance from the line charge to where the field is measured. We can consider another limit, the


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UF PHY 2049 - Electric Field

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