Parth G | The Simple Rule Obeyed by All Electric Fields - Restrictions on the Field @ParthGChannel | Uploaded 2 years ago | Updated 6 minutes ago
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#electromagnetism #electricfield #maxwell #ad
We can't just make up a vector field and assume such an electric field exists in real life! The components of the field in any system have to follow a simple rule in terms of how they relate to each other.
In this video, we start by looking at what is meant by an electric field. We see how electrically charged particles and objects can generate electric fields. They represent what happens when a small positive charge is placed close to the object generating the field. For example, if the field is generated by a negative charge, then the small positive charge will be attracted to it. The closer the two charges, the stronger the attraction.
The electric field of any system / object can be thought of as a vector field, meaning it can be represented by a vector at every point in space. The size of the vector indicates the size of the force exerted on the small positive charge, and the direction indicates which way the force will act.
This means we can visually represent an electric field with either a bunch of arrows, or a set of column vectors (with 3 spatial components) at every point in space. This is important to understand, because each component of the electric field vector can depend on the spatial position that corresponds to that vector, as well as any of the other directions! In other words, the x-component of the electric field, which represents the amount of the electric field pointing in the x-direction at a particular point in space, can depend on the x, y, AND z positions of that point in space.
This becomes important for us when looking at the "simple rule" in question. To understand this rule, we first look at one of Maxwell's Equations of Electromagnetism. It's the one that describes the curl of any electric field as being the negative time rate of change of any magnetic field in the same region of space.
For simplicity, in this video we study systems where the right hand side of the equation, looking at the time rate of change of the magnetic field, is zero. This means we are studying systems with a constant (or zero) magnetic field. We can equate the components of the vector on the left hand side (curl of E) with the zero vector components on the right.
One of the components of curl(E) is formed of the rate of change of an E-field component with respect to another direction, minus the rate of change of the second E-field component with respect to the first direction. For example, the x-component of the curl of E is given by dE_z/dy - dE_y/dz. Since each of these components is equal to zero in our scenario, we can set each of these derivatives to be equal to each other.
What this tells us is that the electric field behaves in a very specific way. The rate of change of the y-component of the field, as we move along z, MUST be equal to the rate of change of the z-component as we move along y. And similar relations exist for the x-z components, and x-y components.
This means real electric fields have a big constraint on how they can behave. We cannot just make up a vector field and expect to find a real field like that in the universe. Even a changing magnetic field changes the restriction on the E-field components, but does not lift it.
The physical significance of this is to do with electric potential difference, or voltage as we discuss it in the study of electric circuits. When a particle moves around in an electric field, and then returns to its original position, the potential difference must be zero (as it returns to the same potential). This is why the RHS of our Maxwell equation became zero.
Videos linked in the cards for this video:
Electric Fields: https://www.youtube.com/watch?v=gSI3PuHQO9A
Maxwell's Equation: https://www.youtube.com/watch?v=6Aab3k2nsOY&t=287s
Curl (Nabla/Del Operator): https://www.youtube.com/watch?v=hI4yTE8WT88
Many of you have asked about the stuff I use to make my videos, so I'm posting some affiliate links here! I make a small commission if you make a purchase through these links.
A Quantum Physics Book I Enjoy: https://amzn.to/3sxLlgL
My Camera (Sony A6400): https://amzn.to/2SjZzWq
ND Filter: https://amzn.to/3qoGwHk
Microphone and Stand (Fifine): https://amzn.to/2OwyWvt
Gorillapod Tripod: https://amzn.to/3wQ0L2Q
Timestamps:
0:00 - What even is an Electric Field?
1:50 - Vector Fields and how to represent them in component form
3:13 - Electric Field components can depend on ANY positional coordinate
4:56 - Maxwell's Equation
6:50 - The codependence of E-field components (when B is constant)
8:43 - The Special Rule!
10:05 - The physical significance of this rule
11:02 - Special shoutout to Squarespace for sponsoring this video!
11:59 - Let me know what to discuss in future videos :)
#ad - This video was sponsored by Squarespace.
Go to Squarespace.com for a free trial, and when youre ready to launch, go to http://www.squarespace.com/parthg to save 10% off your first purchase of a website or domain.
#electromagnetism #electricfield #maxwell #ad
We can't just make up a vector field and assume such an electric field exists in real life! The components of the field in any system have to follow a simple rule in terms of how they relate to each other.
In this video, we start by looking at what is meant by an electric field. We see how electrically charged particles and objects can generate electric fields. They represent what happens when a small positive charge is placed close to the object generating the field. For example, if the field is generated by a negative charge, then the small positive charge will be attracted to it. The closer the two charges, the stronger the attraction.
The electric field of any system / object can be thought of as a vector field, meaning it can be represented by a vector at every point in space. The size of the vector indicates the size of the force exerted on the small positive charge, and the direction indicates which way the force will act.
This means we can visually represent an electric field with either a bunch of arrows, or a set of column vectors (with 3 spatial components) at every point in space. This is important to understand, because each component of the electric field vector can depend on the spatial position that corresponds to that vector, as well as any of the other directions! In other words, the x-component of the electric field, which represents the amount of the electric field pointing in the x-direction at a particular point in space, can depend on the x, y, AND z positions of that point in space.
This becomes important for us when looking at the "simple rule" in question. To understand this rule, we first look at one of Maxwell's Equations of Electromagnetism. It's the one that describes the curl of any electric field as being the negative time rate of change of any magnetic field in the same region of space.
For simplicity, in this video we study systems where the right hand side of the equation, looking at the time rate of change of the magnetic field, is zero. This means we are studying systems with a constant (or zero) magnetic field. We can equate the components of the vector on the left hand side (curl of E) with the zero vector components on the right.
One of the components of curl(E) is formed of the rate of change of an E-field component with respect to another direction, minus the rate of change of the second E-field component with respect to the first direction. For example, the x-component of the curl of E is given by dE_z/dy - dE_y/dz. Since each of these components is equal to zero in our scenario, we can set each of these derivatives to be equal to each other.
What this tells us is that the electric field behaves in a very specific way. The rate of change of the y-component of the field, as we move along z, MUST be equal to the rate of change of the z-component as we move along y. And similar relations exist for the x-z components, and x-y components.
This means real electric fields have a big constraint on how they can behave. We cannot just make up a vector field and expect to find a real field like that in the universe. Even a changing magnetic field changes the restriction on the E-field components, but does not lift it.
The physical significance of this is to do with electric potential difference, or voltage as we discuss it in the study of electric circuits. When a particle moves around in an electric field, and then returns to its original position, the potential difference must be zero (as it returns to the same potential). This is why the RHS of our Maxwell equation became zero.
Videos linked in the cards for this video:
Electric Fields: https://www.youtube.com/watch?v=gSI3PuHQO9A
Maxwell's Equation: https://www.youtube.com/watch?v=6Aab3k2nsOY&t=287s
Curl (Nabla/Del Operator): https://www.youtube.com/watch?v=hI4yTE8WT88
Many of you have asked about the stuff I use to make my videos, so I'm posting some affiliate links here! I make a small commission if you make a purchase through these links.
A Quantum Physics Book I Enjoy: https://amzn.to/3sxLlgL
My Camera (Sony A6400): https://amzn.to/2SjZzWq
ND Filter: https://amzn.to/3qoGwHk
Microphone and Stand (Fifine): https://amzn.to/2OwyWvt
Gorillapod Tripod: https://amzn.to/3wQ0L2Q
Timestamps:
0:00 - What even is an Electric Field?
1:50 - Vector Fields and how to represent them in component form
3:13 - Electric Field components can depend on ANY positional coordinate
4:56 - Maxwell's Equation
6:50 - The codependence of E-field components (when B is constant)
8:43 - The Special Rule!
10:05 - The physical significance of this rule
11:02 - Special shoutout to Squarespace for sponsoring this video!
11:59 - Let me know what to discuss in future videos :)
#ad - This video was sponsored by Squarespace.
