We have spent the last three weeks talking about what happens when you have a changing magnetic field. We found that this changing magnetic field creates a curly electric field. A changing magnetic field then became another source of electric fields. You may then be wondering what happens if you have a changing electric field? We have already seen through Faraday's Law that electric and magnetic fields are related, so how do we account for a changing electric field? Perhaps unsurprisingly, **a changing electric field is another source of curly magnetic fields**. These notes will talk about how we amend Ampere's Law to account for a changing electric field.
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We have spent the last two weeks talking about what happens when you have a changing magnetic field. We found that this changing magnetic field creates a curly electric field. A changing magnetic field then became another source of electric fields. You may then be wondering what happens if you have a changing electric field? We have already seen through Faraday's Law that electric and magnetic fields are related, so how do we account for a changing electric field? Perhaps unsurprisingly, **a changing electric field is another source of curly magnetic fields**. These notes will talk about how we amend Ampere's Law to account for a changing electric field.
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==== Extra Term to Ampere's Law ====
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===== Extra Term to Ampere's Law =====
From Faraday's Law, we learned that a changing magnetic field creates a curly electric field. As a similar parallel, we are now saying that a changing electric field (with time) creates a curly magnetic field. Conveniently, we already have an equation that describes a curly magnetic field: Ampere's Law.
From Faraday's Law, we learned that a changing magnetic field creates a curly electric field. As a similar parallel, we are now saying that a changing electric field (with time) creates a curly magnetic field. Conveniently, we already have an equation that describes a curly magnetic field: Ampere's Law.
∫→B∙d→l=μ0Ienc
∫→B∙d→l=μ0Ienc
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If you remember from a couple of weeks before, Ampere's law says that a current (the Ienc part) will create a curly magnetic field ( the ∫→B∙d→l part). Rather than create a new equation to describe the curly magnetic field from a changing electric field, we instead just add on a term to Ampere's Law:
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If you remember from a couple of weeks before, [[184_notes:motiv_amp_law|Ampere's law]] says that a current (the Ienc part) will create a curly magnetic field ( the ∫→B∙d→l part). Rather than create a new equation to describe the curly magnetic field from a changing electric field, we instead just add on a term to Ampere's Law:
∫→B∙d→l=μ0Ienc+μ0ϵ0dΦEdt
∫→B∙d→l=μ0Ienc+μ0ϵ0dΦEdt
where μ0 is the same constant that we have been dealing with from the last few weeks (μ0=4π⋅10−7TmA),ϵ0 is the same constant from the first few weeks of the semster (ϵ0=8.85⋅10−12C2Nm2), and dΦEdt is the change in //electric// flux (through the Amperian Loop).
where μ0 is the same constant that we have been dealing with from the last few weeks (μ0=4π⋅10−7TmA),ϵ0 is the same constant from the first few weeks of the semster (ϵ0=8.85⋅10−12C2Nm2), and dΦEdt is the change in //electric// flux (through the Amperian Loop).
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This term that we added to Ampere's Law functions in much the same way as Faraday's law. If we can calculate the changing electric flux through a loop, then we can use that to find the magnetic field that curls around that loop. In the example below, we use a charging capacitor to illustrate how this can done; however, for the purposes of this class, we will primarily rely on this idea conceptually (rather than asking you to calculate the magnetic field from a changing electric flux).
This term that we added to Ampere's Law functions in much the same way as Faraday's law. If we can calculate the changing electric flux through a loop, then we can use that to find the magnetic field that curls around that loop. In the example below, we use a charging capacitor to illustrate how this can done; however, for the purposes of this class, we will primarily rely on this idea conceptually (rather than asking you to calculate the magnetic field from a changing electric flux).
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==== Why this Matters ====
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===== Why this Matters =====
With this final piece of the puzzle, we can actually say something really important about how electric and magnetic fields work. If we //__assume that there are no current-carrying wires nearby__//, then we have a set of two equations that say that:
With this final piece of the puzzle, we can actually say something really important about how electric and magnetic fields work. If we //__assume that there are no current-carrying wires nearby__//, then we have a set of two equations that say that:
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==== Examples ====
==== Examples ====
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* [[:184_notes:examples:Week14_b_field_capacitor|Challenge: Magnetic Field from a Charging Capacitor]]
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* Video Example: Magnetic Field from a Charging Capacitor