Engines for electric cars, magnetic resonance imaging in medicine,
electric kettles in the kitchen, the charger of your smart phone, radio, wifi
and so on. Any device that exploits electricity or magnetism is fundamentally
based on the Maxwell equations. The goal of this video is not to derive the
Maxwell equations theoretically or experimentally, but to present them as
simple and understandable as possible. I will also briefly explain the
mathematics that occur in the Maxwell equations. However, you should know
what a partial derivative is and what an integral is before you continue
to watch the video. To understand all Maxwell equations you first need
to know what an electric and magnetic field is. Consider a large electrically
charged sphere with the charge ‘capital Q’ and a small sphere with the
charge ‘small q’. The electric force between these two spheres, which are at
a distance r from each other, is given by Coulombs law. Here ‘one over 4 Pi
epsilon zero’ is a constant prefactor with the electric field constant ‘epsilon
zero’, which provides the right unit of force, namely the unit Newton.
Now what if you know the value of the big charge and want to know
the value of the force that the big charge exerts on the small charge?
However, you do not know the exact value of this little charge, or you
intentionally leave that value open and only want to look at the electric force
exerted by the big charge. So you have to somehow eliminate the small
charge q from Coulombs law. To do so, you simply divide Coulombs law by q
on both sides. This way, on the right hand side, the small charge drops out
and instead lands on the
left hand side of the equation. The quotient on the left hand side ‘F
over q’ is defined as the electric field E of the source charge Q. By calling it
source charge we imply that the charge Q is the source of the electric field.
The electric field E thus indicates the electric force which would act on a
small charge when placed at the distance r to the source charge. So far, only
the magnitude, that is, the strength of the electric field has been considered,
without taking into account the exact direction of the electric field.
However, the Maxwell equations are general and also include the
direction of the electric field. So we have to turn the electric field into a
, vector. Vectors are shown in boldface. Handwritten mostly with a little arrow
above the letter to distinguish it from scalar (pure) numbers. I omit the
arrows because they make the equations unnecessarily ugly. The electric
field E as a vector in three-dimensional space has three components Ex, Ey
and Ez. Let’s look at the first component. The first component depends on
the space coordinates (x, y, z) and is the magnitude of the electric
field in x-direction. That is, depending on which concrete location is
used for (x, y, z), the value Ex is different. The same applies to the other two
components Ey which indicates the magnitude in the y direction and Ez
which indicates the magnitude in the z direction. The components of the
electric field thus indicate which electric force would act on a test charge at a
specific location in the first, second or third spatial direction. Another
important physical quantity found in the Maxwell equations is the magnetic
field
B. Experimentally, it is found that a particle with the electric charge q
moving at the velocity v in an external magnetic field experiences a
magnetic force, which deflects the particle. The force on the particle
increases in proportion to its charge q and its velocity v, that is doubling the
charge or the speed doubles the force on the particle. But not only that! The
force also increases in proportion to the applied magnetic field. To describe
this last proportionality of the force and the magnetic field, we introduce
the physical quantity B. The unit of this quantity must be such that the
right hand side of the equation gives the unit of force, that is, ‘Newton’ or
‘kilogram meter per second sqaured’. By a simple transformation, you will
find that the unit must be ‘kilogram per ampere second squared’. This is
what we will call the unit of Tesla. And we call B magnetic flux density (or
short: magnetic field). The magnetic flux density describes the external
magnetic field and thus determines the magnitude of the force on a charged
particle.
The equation ‘q times v times B’ for the magnetic force on a charged
particle represents only the magnitude of the force. In order to formulate the
magnetic force with vectors – analogous to the electric force – the force, the
electric kettles in the kitchen, the charger of your smart phone, radio, wifi
and so on. Any device that exploits electricity or magnetism is fundamentally
based on the Maxwell equations. The goal of this video is not to derive the
Maxwell equations theoretically or experimentally, but to present them as
simple and understandable as possible. I will also briefly explain the
mathematics that occur in the Maxwell equations. However, you should know
what a partial derivative is and what an integral is before you continue
to watch the video. To understand all Maxwell equations you first need
to know what an electric and magnetic field is. Consider a large electrically
charged sphere with the charge ‘capital Q’ and a small sphere with the
charge ‘small q’. The electric force between these two spheres, which are at
a distance r from each other, is given by Coulombs law. Here ‘one over 4 Pi
epsilon zero’ is a constant prefactor with the electric field constant ‘epsilon
zero’, which provides the right unit of force, namely the unit Newton.
Now what if you know the value of the big charge and want to know
the value of the force that the big charge exerts on the small charge?
However, you do not know the exact value of this little charge, or you
intentionally leave that value open and only want to look at the electric force
exerted by the big charge. So you have to somehow eliminate the small
charge q from Coulombs law. To do so, you simply divide Coulombs law by q
on both sides. This way, on the right hand side, the small charge drops out
and instead lands on the
left hand side of the equation. The quotient on the left hand side ‘F
over q’ is defined as the electric field E of the source charge Q. By calling it
source charge we imply that the charge Q is the source of the electric field.
The electric field E thus indicates the electric force which would act on a
small charge when placed at the distance r to the source charge. So far, only
the magnitude, that is, the strength of the electric field has been considered,
without taking into account the exact direction of the electric field.
However, the Maxwell equations are general and also include the
direction of the electric field. So we have to turn the electric field into a
, vector. Vectors are shown in boldface. Handwritten mostly with a little arrow
above the letter to distinguish it from scalar (pure) numbers. I omit the
arrows because they make the equations unnecessarily ugly. The electric
field E as a vector in three-dimensional space has three components Ex, Ey
and Ez. Let’s look at the first component. The first component depends on
the space coordinates (x, y, z) and is the magnitude of the electric
field in x-direction. That is, depending on which concrete location is
used for (x, y, z), the value Ex is different. The same applies to the other two
components Ey which indicates the magnitude in the y direction and Ez
which indicates the magnitude in the z direction. The components of the
electric field thus indicate which electric force would act on a test charge at a
specific location in the first, second or third spatial direction. Another
important physical quantity found in the Maxwell equations is the magnetic
field
B. Experimentally, it is found that a particle with the electric charge q
moving at the velocity v in an external magnetic field experiences a
magnetic force, which deflects the particle. The force on the particle
increases in proportion to its charge q and its velocity v, that is doubling the
charge or the speed doubles the force on the particle. But not only that! The
force also increases in proportion to the applied magnetic field. To describe
this last proportionality of the force and the magnetic field, we introduce
the physical quantity B. The unit of this quantity must be such that the
right hand side of the equation gives the unit of force, that is, ‘Newton’ or
‘kilogram meter per second sqaured’. By a simple transformation, you will
find that the unit must be ‘kilogram per ampere second squared’. This is
what we will call the unit of Tesla. And we call B magnetic flux density (or
short: magnetic field). The magnetic flux density describes the external
magnetic field and thus determines the magnitude of the force on a charged
particle.
The equation ‘q times v times B’ for the magnetic force on a charged
particle represents only the magnitude of the force. In order to formulate the
magnetic force with vectors – analogous to the electric force – the force, the
